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P. J. HALER, M.B.E., B.Sc., MEM.AM.Soc.M.E., 

A.M.I.MECH.E., A.I.E.E. 






A. H. STUART, B.Sc., F.R.A.S. 






TTfoe niibrars press Olimitefc 




PHYSICS is generally acknowledged to be essential to the 
equipment of every technical student, and the subject forms 
part of the curriculum of most Junior Day Technical Schools 
and Trade Schools. 

In the following pages a scheme of the subject is developed, 
which forms a two-year course when two or three hours a 
week are devoted to the subject. 

Since most technical students take Applied Mechanics 
and Electricity as separate subjects, the portions of Physics 
usually treated under .these heads have been purposely 
omitted from this course and thus the danger of overlapping 
has been avoided. 

Apart from these omissions the whole of the elements of 
the subject have been passed under review, and if the treat- 
ment of certain portions is brief, it is hoped that the breadth 
of outlook which the student derives from such a running 
survey will be considered sufficient compensation. 

The experimental work has been confined to such as 
requires only the very simplest apparatus. Home-made 
appliances have a very distinct advantage over the more 
professional type at this stage of the student's work. 

Sections I., II., and III. form a suitable course for the 
first year, and Sections IV. and V. may be completed during 
the second year. 

As one of the authors is in the employment of the London 
County Council it is necessary in accordance with regulations 
to state that the London County Council is in no way re- 
sponsible for the contents of this book. 


August, 1921. A. H. S. 


MANY students of all ages and of all types show a marked 
tendency to regard certain subjects as being "useful," and 
others as being the reverse. No policy could have a more 
sinister influence on progress or place such a final limit on 
the student's actual usefulness in his vocation. 

If any of the great discoveries or inventions which have 
revolutionised civilisation be examined and tracked back 
to its source, we invariably find a man working in some 
apparently quite useless field of research, with no object in 
view other than the acquisition of knowledge. Utility never 
enters his head, for the stage of the work engaging his immedi- 
ate attention is such that no living man is in a position to 
say what is " useful " and what is not. The useless of to-day 
may be of paramount importance to-morrow. 

When Volta in 1800 made his voltaic pile and obtained 
a feeble current of electricity, and Oersted in 1820 discovered 
the action of a current on a magnet ; when Davy in 1821 
demonstrated the power of a current to magnetise steel, and 
Faraday in 1831 showed that a current in one circuit could 
induce a current in another circuit, no one could foretell 
that these discoveries would lead to the production of a 
dynamo which, when rotated at Chelsea, could propel an 
electric train at Hampstead. 

The contemporaries of the pioneers just mentioned may 
have said to them : " This is all very interesting, but what 
is the use of it ? " If they had answered : " In less than a 
century these principles will enable a man in London to 
speak to a man in Paris," they would have been laughed at. 

Again, we may note that great discoveries and inventions 
have seldom if ever been made by one man. They are all 


the results of the cumulative efforts of many workers. We 
generally associate the steam turbine with the name of 
Parsons, yet the turbine was only made possible by the 
development of the thermo-dynamics of steam carried out 
by Kelvin, Rankin, and others. 

The course of Physics which is expanded in the following 
pages is intended for technical students, chiefly those associ- 
ated with engineering. Yet its object is not to teach engi- 
neering ; it is not primarily to teach physics. The object 
which Ihe authors have had constantly in mind is to give 
to the student the outlook of a physicist upon engineering. 

Education in any sphere does not consist in committing 
a number of facts to memory : it aims at producing an outlook 
on life or an attitude of mind. The mental attitude of a 
physicist is one of quantitative observation. He looks out 
on Nature and measures what he sees, and as physics deals 
chiefly with the sources of energy, it has, in the past, con- 
tributed much of value to engineering, and doubtless has more 
to offer in the future. 
















IX. PHOTOMETRY . . . . . .104 


XI. REFRACTION . . . . . .121 



XIII. THERMOMETRY ..... . 146 




ANSWERS ....... 233 

INDEX 239 




UNITS - . 15 



THE MUSICAL SCALE . . . . . .81 

REFRACTIVE INDICES . . . . . .127 

TEMPERATURES . . . . . . .157 


SPECIFIC HEATS (GASES) . . . . . .187 







Units. Physics is the science in which the properties of 
matter and the properties of energy are investigated quanti- 
tatively. All measurement involves a comparison between 
the thing to be measured and some standard quantity of the 
same nature, this standard being called the " unit." 

It is very desirable that the number of different units 
employed in our measurements should be as small as possible, 
and it will be a further advantage if new units, as they become 
necessary, are made to depend upon those already in use. 

Fundamental Units. A fundamental unit is one which 
is necessarily independent of the others. For example, 
length is a measurement of a fundamental nature. In this 
country the unit of length defined by Act of Parliament is the 
yard. Multiples or submultiples of this are, of course, em- 
ployed according to the magnitude of the thing to be measured. 
The foot (i.e., one third of a yard) is frequently employed in 
scientific measurement. 

Another unit of length (defined by the law of France) is 
the metre. One hundredth part of this, called a centimetre, 
is the unit of length adopted in many physical measurements. 

Mass, the amount of matter in a body, is another funda- 
mental unit. The units of mass commonly used are the 
pound (in the British system) and the gramme (in the metric 

Time is another fundamental idea of which we require 
a unit. In physics the second, which is 8 ei 00 * a mean s l ar 
day, is universally used. 



Derived Units. The units just described deal with three 
quite independent ideas. If, however, we turn to ideas of 
area and volume it is unnecessary to define new independent 
units, for these quantities can be measured by the repeated 
determination of length. Thus for area we naturally use 
the square foot or the square centimetre as a unit, while the 
cubic foot or the cubic centimetre is a suitable unit of volume. 
These are called derived units. 

The^e are a great many derived units, for experience has 
shown that almost anything may be measured in units 
derived from the fundamental units of length, mass, and 

These derived units may be divided into two classes. 

(1) Those using the foot, the pound, and the second, 

belong to the " F.P.S. system." 

(2) Those using the centimetre, the gramme, and the second, 

belong to the C.G.S. "system." 

Weight and Mass. If we take a 1 Ib. weight we may say 
that it contains one pound of matter. If we hang it on a 
spring balance the latter records the fact that this piece of 
matter " weighs " one pound. 

Suppose we take the arrangement on to a lift. As the 
lift commences to rise the recorded weight of the body would 
be more than 1 Ib., while as the lift was about to stop again 
the balance would show a weight of less than 1 Ib. 

Again, if we carried the balance and weight about the surface 
of the earth, we should have recorded more than 1 Ib. near 
the poles of the earth, and less than 1 Ib. near the equator. 

Or -yet another case. If the test were made on a high 
mountain or at the bottom of a deep pit, the balance would 
indicate a weight than less 1 Ib. 

It appears therefore that the weight of a body may vary 
from place to place, since it depends upon the attraction of 
the earth, which is known to vary in intensity. But it is 
contrary to sense for the amount of matter (or mass) to vary. 

In using 1 Ib. or 1 gramme as the unit of mass, the unit is 
defined as the weight of a standard at a certain place and at a 
certain level, such as sea level at Greenwich. 


Engineers sometimes find it convenient to use the pound 
as the unit of weight and g Ibs. as the unit of mass, g (the 
acceleration caused by the earth's gravity) being defined 

Velocity. Velocity may be defined as the rate of change 
of position. The units commonly employed are one foot 
per second, and one centimetre per second. A velocity of 
1 foot per second indicates that the body is increasing its 
distance from some point of reference at the rate of one foot 
in every second. 

Acceleration. Acceleration is the rate of change of velocity. 
Thus, if a body at a particular instant of time is moving 
with a velocity of 5 feet per second, and after a period of 
4 seconds its velocity has increased to 17 feet per second, 
the amount of change of velocity is 12 feet per second, and 
this change has taken place in 4 seconds. The change has, 
therefore, been at the average rate of 3 feet per second each 
second. This is usually written : 3 feet per second per 

The units of acceleration commonly employed in physics 
are the foot per second per second, and the centimetre per 
second per second. 

If a body be allowed to fall freely to the earth, its velocity 
constantly increases during its fall, owing to the persistent 
attraction which the earth exerts. Experiment shows that, 
falling freely from rest, a body acquires a velocity of about 
32 -2 ft. per second in the first second, and at the end of two 
seconds its velocity has become 64*4 ft. per second, and so on. 
Hence we see that the acceleration is at the rate of 32 -2 ft. 
per second per second. 

This value is generally denoted by the letter g. Measured 
in C.G.S. units its value is 981 centimetres per second per 

Force. Force is that which tends to overcome inertia. 
If one pushes or pulls at an obstacle with a force equal to 
the weight of one pound, a force is exerted, but the resistance 
to motion may or may not be overcome. The unit of force 


is sometimes a force equal to the weight of one pound (or 
one gramme). 

In many cases, however, it is convenient to define the 
unit of force as that which, acting on unit mass for unit 
time, gives it unit velocity. From this definition it is easy 
to see that F=Ma, where F is the force which, acting on M 
units of mass, produces in it a units of acceleration. 

In the F.P.S. system this unit is called the poundal, and 
in the C.G.S. system it is called the dyne. It follows that 
there are g poundals in a force equal to one pound, and g 
dynes in a force equal to one gramme. For : 

F (in poundals) =M (in lbs.)xa (in ft. per sec. per sec.), 
also F (in dynes) =M (in grammes) X a (in cms. per sec. per 

Work. Work is done when resistance is overcome. The 
unit of work is done when unit force acts through unit distance. 
Thus the foot-pound is the work done by a force of one pound 
acting through one foot, and the foot-poundal is the work 
done by a force of one poundal acting through one foot. 

An erg is the C.G.S. unit of work, and is the work done by 
a force of one dyne acting through a distance of one centi- 

Energy. Energy is that which is capable of doing work. 
Thus a rotating flywheel possesses kinetic energy. A weight 
raised above the earth's surface possesses potential energy 
in virtue of its position, and this energy is converted into 
kinetic energy when the body is allowed to fall to earth. 

Heat is a form of energy which is converted into kinetic 
energy by a steam or internal combustion engine, and electri- 
city is another form of energy which may be converted into 
heat by a resistance or into kinetic energy by an electric 
motor. Coal and petrol both possess chemical energy, which 
is converted into heat by combustion. 

$IEnergy may be measured in the same units as work, since 
the* energy of a body is expressed by the amount of work it 
can do. 



Units of the Units of the 

C.G.S. System. F.P.S. System. 

Length . . Centimetre. Foot. 

Mass . . Gramme. Pound. 

Time . . Second. Second. 

Area . . Sq. cm. Sq. ft. 

Volume . . Cub. cm. Cub. ft. 

Velocity . . Cm. per sec. Ft. per sec. 

Acceleration . Cm. per sec. per sec. Ft. per sec. per sec. 

Force . . Dyne. Poundal. 

Work . . Erg. Ft.-poundal. 

Energy . . Erg. Ft.-poundal. 

N.B. The poundal and the foot-poundal are the true 
F.P.S. units of force and work respectively, although the 
pound and the foot-pound are often used by engineers. 


Length . 
Volume . 

1 ft. =30-48 cms 
1 Ib. =453-6 gms. 
1 sq. ft. =929 sq. cms. 
1 cu. ft. =2-83xl0 4 c.c. 

Velocity . 

1 ft. per sec. =30*48 cms. 
1 ft. per sec. per sec. =30*48 cms. 

per sec. 
per sec. 

Work and energy 

1 pdl. =1 -38 X10 4 dynes. 
1 ft.-pdl. =4*21xl0 5 ergs. 

per sec 

1 lb.=0 pdls.=4-45x 10 5 dynes. 
1 ft. - \b.=g ft.-pdls.=l-236x 10 ergs. 


Ex. 1. Express a velocity of 50 miles per hour in feet per 
second and cms. per second. 

__50 miles 
1 hour. 
^50x1760x3 ft. 

60x60 sees. 
=73*3 ft. per sec. 

Again 73*3 ft. ^73 -3x30 '48 cms. 
1 sec. 1 sec. 

=2235 cms. per sec. 

Ex. 2. At a given instant a train is travelling with a 
velocity of 24 miles per hour. Five minutes later the velocity 
is 46 miles per hour. Express the acceleration in (a) miles 
per hour per hour, (b) feet per second per second, (c) cms. per 
second per second. 

The train gains 22 miles per hour in one-twentieth of an 
hour. The average acceleration is therefore 440 miles per 
hour per hour. 

The student should note that while it is possible for a 
train to accelerate at this rate for a few minutes, the influence 
of friction is such that the acceleration cannot be maintaned 
for anything like an hour and hence the train never attains 
a velocity of 440 miles per hour. 

440 miles 

Now acceleration -JT^: -- ^2 
(1 hour) 2 

_ 440x1760x3 ft. 
(60x60 sees.) 2 
=0*179 ft. per sec. per sec. 
Also 0-179 ft. _0-179x 30-48 cms. 
"(1 sec.) 2 (1 sec.) 2 

5*45 cms. per sec, per sec. 


Ex. 3. The average pressure of the atmosphere at sea 
level is 14*7 Ibs. per sq. in. Express this in dynes per sq. 

14-7 Ibs. 

Pressure = -n~ ^ 9 
(1 m.) 2 

14-7xgXl-38xlQ 4 dynes 

(2-54 cms,) 2 
_14-7 X 32-2x1 '38 xlO 4 dynes 

6*45 sq. cms. 

=l'014x 10 dynes per sq. cm. 
Or approximately a million dynes per square centimetre. 

Ex. 4. A weight of 7 Ibs. is raised vertically through 4 ft. 
Express the work done in foot-pounds, foot-poundals, and 

Work done =1 Ibs. X 4 ft. 
=28 ft.-lbs. 
=901 ft.-pdls. 
=901x4-21 XlO 5 ergs. 
=3 '8 XlO 8 ergs. 

Exercises 1. 

1. Convert the following measurements from inches to 
millimetres : 1.^, 2 T V, 3 T y. Check the results graphically. 

2. Convert the following measurements from millimetres 
to inches : 3 '175, 4 '762, 307 '97. Check the results graphic- 

3. Convert the following measurements from metres to 
feet: 1-1, 1-5, 3 '9. 

4. Convert the following areas from square centimetres to 
square inches : 1'2, 2 '6, 9 '7. Check the results graphically. 

5. If 1 pound =0*45359 kilogramme, convert the follow- 
ing readings in pounds to kilogrammes : I'l, 2 '5, and 4 '7. 


6. If 1 kilogramme = 2 '2046 pounds, convert the following 
readings from kilogrammes to pounds : 5*1, 6 '5, 7*7. 

7. How many poundals are represented by the following 
pounds weight : 1'5, 2 '8, and 3 '7 ? 

8. Express a velocity of 60 miles per hour in feet per second 
and centimetres per second. 

9. A 'bus is travelling at the rate of 12 miles per hour. 
What is this velocity in feet per minute ? 

10. Observations were taken to determine the velocity of 
a part of a mechanism and the following results were obtained : 

Time in minutes . . .012345 
Space passed over in inches .0 3 6 9 12 15 

Plot a space -time graph. If space divided by time = velocity 
(when the velocity is uniform), determine the velocity for 
each minute, and plot a velocity- time graph. 

11. The previous experiment was repeated on another piece 
of mechanism and results were obtained as follows : 

Time intervals in minutes .1 23 4 

Space passed over in inches . 3'65 7 '30 10'95 14'6 

Plot a graph of time and space. What is the velocity in 
feet per minute for every minute ? Plot a graph of velocity 
and time. 

12. Express a velocity of 30 miles per hour in feet per minute 
and centimetres per minute. 

13. A wheel of 4 ft. 3 ins. diameter makes 200 revolutions 
per minute. What is the velocity of a point on the rim, in 
feet per second ? 

14. At a given instant a train is travelling at the rate of 
30 miles per hour and 5 minutes later the velocity is 35 
miles per hour. Express the average acceleration (a) in 
miles per hour per hour, (6) feet per second per second, (c) 
centimetres per second per second. 

15. A piece of steel is stressed to 2 tons per square inch. 
Express this stress in dynes per square centimetre. 



16. The pressure in a boiler is 100 Ibs. per square inch. 
Express this in dynes per square centimetre. 

17. A weight of 4 pounds is raised through a vertical height 
of 8 feet. Express the work done in foot-pounds, foot-poundals 
and ergs. 

18. It is noted that in 65 seconds a car has travelled one 
kilometre. Express the velocity in kilometres per hour and 
miles per hour. 



Matter. Matter is generally considered as that which 
possesses weight. It is conveniently divided into three 
classes, known as solids, liquids, and gases. 

Solids. A solid is that form of matter which may be 
submitted to compression without lateral support. Thus, iron 
being a solid, it is possible to take an iron cylinder, stand it 
on its base and place a weight on the top. The iron is now 
in compression, and, provided the weight is not excessive in 
relation to the dimensions of the cylinder, no appreciable 
deformation is produced. 

Here we have a solid in compression without lateral sup- 
port. Water, on the other hand, being a liquid, could not be 
treated thus. One cannot have a " cylinder of water " unless 
the water is supported by a vessel of some solid material. 
If we wish to compress a liquid or a gas it must be supported 
in a tube or other vessel. 

Light and Heavy Solids. Aluminium is spoken of as a 
"light" metal, and lead is said to be "heavy." Both these 
expressions are intended to give an indication of the weight 
of a piece of matter in relation to its volume. For a pound of 
aluminium weighs as much as a pound of lead, but the former 
occupies nearly 4| times the volume of the latter. 

Specific Gravity. The specific gravity of a body is its 
weight compared with the weight of an equal volume of 
something else. Thus it would be legitimate to say that 
the specific gravity of lead is 4|, meaning that it is 4J times 
as heavy as an equal volume of aluminium. But if it were 
compared with iron, the specific gravity would be only about 
1J, whereas compared with platinum it would be about J. 




For the numerical value of the specific gravity of a body 

to be of any practical use, it is necessary to have some standard 

substance with which all other bodies may be compared. 

Water has been selected for this purpose and hence we have : 

The Specific Gravity of a body is the weight of that body 

compared with the weight of an equal volume of water. 

It will be seen in Chapter XV that this definition, to be 
complete, must state the temperature of the water (which 
should be at 4 C.), but the student may neglect this point 
at the present stage of his work. 

Density. The Density of a substance is the weight of unit 
volume of that substance. 

It will be seen that whereas the specific gravity of a sub- 
stance has a constant numerical value, the Density will 
vary with different units. Thus the specific gravity of 
copper is 8*93, which means that a piece of copper weighs 
8 '93 times as much as an equal volume of water. 

The Density of copper is 555-lbs. per cub. ft. and 0'321-lb. 
per cub. in., and 8 '93 grammes per c.c. We might add to these 






Length of edge =1. 


Area of face = A. 


Length =Z. 



' Area of Base = A.' 
Vertical height =h. 

I Ah 


' Diameter f =d. 
Length =1. 



' Diameter of base =d. 
Vertical height =h. 


Sphere . 

Diameter =d. 



by changing the units. The student should note carefully 
that specific gravities may be expressed by mere numbers 
without the mention of any units, but in expressing a density 
the units employed must be stated. 

From the foregoing it will be clear that in the practical 
determination of specific gravities and densities it is only 
necessary to find the volume and weight of a specimen 
of the material. If the specimen is a regular solid such as 
a cube or a cylinder, the volume may be calculated from one 
or two simple measurements. 

The table on p. 21 will be of use to those who are not yet 
familiar with the methods employed in finding the volumes 
of regular solids. 

Determination of Weight. The weight of a body may be 
determined in two ways : 

(1) By noting the deformation of a spiral spring when the 

latter is put into tension or compression by the body 
in question. 

(2) By balancing it with standard weights. 

In the first case the spring has usually been calibrated, 
and the weight is recorded by a needle passing over a gradu- 
ated dial. 

In the second case a pair of scales and a set of weights may 
be used, similar to those employed in many shops. 

In either case the weight recorded is only an approximation 
and such methods should only be employed when a rough 
determination is required on a comparatively large quantity 
of matter, say a pound or two. For more accurate work a 
chemical balance should be used. 

The Balance. This instrument is similar in general prin- 
ciple to that of a pair of scales, but a high degree of precision 
is maintained in every detail of it. Gramme weights are gener- 
ally employed in connection with a balance of this nature. 
The student should examine carefully a balance and a box 
of gramme weights and note their arrangement. In using a 
balance it is necessary to observe certain rules, as a balance 
is such a delicate piece of mechanism that it is easily damaged. 



1. See that the box of weights is complete. 

2. See that the base of the balance is level as indicated 
by the spirit level or plumb bob. If not, adjust the levelling 

3. Never place anything on a pan or remove anything 
from it unless the beam is resting on its supports. 

4. When operating the lever which raises the beam from 
its supports never make a " jerky " movement. Endeavour 
to bring the beam in contact with its supports when the 
needle indicates that the beam is horizontal. 

5. Raise the beam and observe the swing of the needle. 
It should swing an equal number of divisions of the scale 
on either side of the centre mark. If it does not do this the 
balance needs adjusting by means of the screw provided 
for the purpose. Students should, however, remember that 
this adjustment should not frequently be necessary, and in 
any case it should only be made by a person of experience. 

6. No substance which is at all likely to injure the balance 
pan should ever be placed on the bare pan. Such substance 
should be weighed on a watch glass of known weight. 

7. It is convenient to place the body to be weighed on the 
left-hand pan of the balance. The weights can then be operated 
by means of the forceps in the right hand, while the left 
hand operates the lever which raises the beam. 

8. Don't guess at suitable weights. Find one which is too 
heavy, after which every weight below this one should be 
placed, separately, on the pan and the beam raised. The 
weight is allowed to remain if the weight total is too small, 
and removed to the box if the weight total is too great. 

9. Every weight should be either on the pan or in the box. 

10. Never put weights on the pan carrying the body to be 
weighed. It is quite unnecessary. 

11. The' weights are correct when the needle swings an 
equal number of divisions on either side of the centre mark. 
Don't wait for the beam to come to rest. 

12. Never leave the beam unsupported longer than is 
absolutely necessary. 


13. Determine the total of the weights by examining the 
spaces in the box. 

14. Replace the weights in the box, totalling them as you 
do so. The totals should of course agree. 

15. Every balance is designed to carry not more than a 
certain load on each pan. This is often 250 or 500 grammes. 
Great care must be taken never to over-load a balance. 
Carelessness on this point may cause permanent injury to 
the balance. 

Ex. 5. A solid cylinder of brass is 3*5 cms. long, and has 
a diameter of 1'37 cms. It weighs 43 '873 grams. Find its 
density in grams per c.c. and in Ibs. per cubic in. Also find 
its specific gravity. 

Area of circular base = '7854 (1'37) 2 sq. cms. 
Volume of cylinder = -7854 (I'37) 2 x3'5 c.c. 

Weight 43'873 

Volume ~ '7854 (I'37) 2 x3'5 
= 8*5 grams per c.c. 
8*5 grams 

Density = 

1 c.c. 





8-5 X 16-39 

cub. in. 

Ibs. per cub. in. 

= '307 Ibs. per cub. in. 

Since 1 c.c. of water weighs 1 gramme it follows that the 
specific gravity of any body is numerically equal to the 
density expressed in grammes per c.c. 

Hence the specific gravity of the brass of which this cylinder 
is made is 8 '5. 


Irregular Solids. The methods adopted for the determina- 
tion of the density of irregular solids will be more conveniently 
considered in Chapter III. 

Experiment 1. 

Determine the density in grammes per c.c., and Ibs. per 
cub. in. of the material composing any regular solids, such 
as cubes, cylinders, spheres, etc., which are available. 

Exercises 2. 

1. Determine the volume of the following cubes, dimensions 
being given in centimetres in each case : 1*9, 2*4, 4*3, 0'45. 

2. Determine the volume of the following cylinders, each 
cylinder being 1" long and the diameters as given : 0*1", 
1-2", 3-3", 4*5". 

3. Determine the volume of the following spheres, the 
diameters being given in inches : 0*1", 0*4", 2 *5", 8 '7". 

4. An indiarubber stopper weighs 26*545 grammes and its 
volume is 18*195 cubic centimetres. Determine its density 
in grammes per cubic centimetre and its specific gravity. 

5. A piece of steel I" wide, \" thick and 12" long weighs 
0'85 Ibs. Determine its density in Ibs. per cubic inch, its 
volume in cubic centimetres, and its weight in grammes. 
What is the specific gravity ? 

6. Determine the density in Ibs. per cubic inch of the follow- 
ing pieces of mild steel. 

Breadth in ins. Thickness in ins. Length in ins. Weight in Ibs. 

1 i 12 0*85 

H -A 12 1*2 

It f 12 1*59 

7. Determine the density of the following pieces of mild 
steel, each piece being square in cross-section: Length of edge, 
iV, i", and -,y. Each piece is 12" long and weighs 0*12, 
0*213, and 0*651 Ibs. respectively. 



8. Find the density of cylindrical bars 12" long and of the 
following diameters and weights, f" dia., 1*912 Ibs. |f" dia., 
2-245 Ibs., and | dia., 2'603 Ibs. 

9. The density of the following substances is given in 
Ibs. per cubic foot. Determine the density in grammes per 
cubic centimetre : Water 62*4, aluminium 161*7, antimony 
417, and bismuth 613. 

10. The mass of a cubic foot of water is nearly 1000 ozs. 
(998*6 ozs.), so it often said that a cubic foot of water weighs 
1000 ozs. A litre of water weighs 1000 grammes. Deter- 
mine the density of water in Ibs. per cubic foot and in grammes 
per cubic centimetre. 

11. The relative density of a substance is the ratio of the 
mass of the body composed of it to the mass of an equal 
volume of the standard substance. For the purposes of this 
question take the standard substance as water at 62 F., 
weighing 62*355 pounds per cubic foot. What is the relative 
density of the following substances, whose weights are given 
in Ibs. per cubic foot ? Aluminium, 160 ; brass, 530 ; 
copper, 555 ; mild steel, 480. What is another name for 
relative density ? 

12. The average weight in Ibs. of one cubic foot of various 
substances is given. Determine the specific gravity in each case. 
Chalk, 156. Flint, 162. Pure cast gold, 1,204. Wrought 
iron, 480. Silver, 655. 

13. A table is given showing the weight per square foot 
for material of the thicknesses shown. Determine the density 
hi Ibs. per cubic inch in every case. 


Weight in Ibs. 
per sq. foot. 

Thickness in ins. 












14. Determine the specific gravity of the following timbers, 
given their weight in Ibs. per cubic foot : 

Alder, 42. Apple, 47. Ash,. 45. Beech, 46. Birch, 41. 

15. The specific gravity of the following liquids taken at 
60 F. is given : Nitric acid, 1 '54. Sulphuric acid, 1 '849. 
Pure alcohol, 0-794. Ammonia (27 '90 per cent.), '891. Deter- 
mine in each case the weight of a cubic foot. 



Fluids. A fluid is the general class name given to any 
form of matter which flows. The class includes gases, vapours, 
and liquids. 

Liquids. A liquid is a form of fluid which has a definite 
volume. It takes the shape of the vessel into which it is 
put, but does not expand to fill the vessel as does a vapour or 
a gas. 

Density Of a Liquid. In determining the density of a liquid 
the method already applied to solids is used. Two measure- 
ments have to be made : 1. The volume of the liquid in 
question must be obtained. 2. The weight of this volume of 
the liquid must be determined. The calculation is then made 
as in deter mining the density of a solid. 

A liquid cannot of course be measured except in a contain- 
ing vessel, and it is necessary that the measurements of 
volume and weight should both be made while 
the liquid is in the same containing vessel, since 
liquid transferred from one vessel to another 
always diminishes in quantity owing to some 
remaining on the " wet " sides of the former 

Specific Gravity Bottle. Fig. 1 shows the 
common form of a specific gravity bottle. It 
consists of a thin glass flask, fitted with a 
ground stopper through which a fine hole has 
been drilled. If the stopper is removed and 
the bottle filled to the brim with a liquid, on 
gently lowering the stopper into position the 


FIG. 1. 



superfluous liquid flows through the fine hole in 
the stopper. The bottle is now exactly " full," 
and if it is made to hold a specified amount 
(usually 25 c.c.) we know the volume without 
further effort. 

If a specific gravity bottle is not available a 
common 1 oz. stoppered bottle forms a good sub- 
stitute, provided the stopper has been grooved 
down its side by means of a triangular file. (See 
Fig. 2.) In this case, however, the bottle must 
be calibrated for volume. An example will make 
this clear. 

FIG. 2. 

Ex. 6. A stoppered bottle weighs, when empty, 16 '835 
grms. Filled with pure cold water it weighs 43 '728 grms. 
Find its capacity. 

Weight of bottle and water . . . =43*728 grms. 
Weight of bottle alone . . . =16-835 

Weight of water alone 


Since 1 c.c. of water weighs 1 grm. the capacity of the 
bottle is 26-89 c.c. 

Ex. 7. Using the bottle mentioned in Example 6, it was 
found that the weight when rilled with lard oil was 41*457 
grms. Find the specific gravity of the oil. 

Weight of bottle and oil 
Weight of bottle alone 

Weight of oil 

41*457 grms. 


Now the result of Example 6 showed that the capacity 
of this bottle was 26 -89 c.c. 


Hence 1 c.c. of lard oil weighs -^ - grins. = 0'916 grms. 

per c.c. 

The specific gravity of lard oil is therefore 0'916. 


Experiment 2. 

Obtain a 1 oz. stoppered bottle and having cut a groove in 
the stopper calibrate it for volume by the method indicated 
in Example 6. If a. specific gravity bottle of the pattern 
shown in Fig. 1 is to be used, it, too, should be calibrated in 
case the volume marked upon it is not quite correct. 

Experiment 3. 

Prepare a saturated solution of common salt in water. 
Find the density of this solution by the method indicated in 
Example 7. 

Experiment 4. 

Take some of the saturated solution of salt already pre- 
pared and mix with it an equal volume of water. Having 
thoroughly mixed these two liquids, find the density of the 
mixture. Is the density the mean of the density of water 
and that of the salt solution ? 

Experiment 5. 

Find the density of alcohol. (If alcohol is not available, 
common methylated spirit may be used. This is an impure 
form of alcohol.) 

Experiment 6. 

Mix alcohol and water together in equal parts and find the 
density of the mixture. Is the density the mean of those of 
alcohol and water ? 

Experiment 7. 

If the results of experiments 4, 5, and 6 appear at all 
confusing, test whether on mixing, say, 50 c.c. of water and 
salt solution, or water and alcohol, 100 c.c. of mixture is 
obtained. It will be a wise precaution to have a thermometer 
in. the liquids to ensure that all measurements are made at 
the same temperature. 

Level. A " level " surface may be defined as one which 
is everywhere at right angles to the direction in which the 
earth's gravitational force acts. Where only small areas 



are being considered a level surface may 
be regarded as being " flat " or " plane," 
but when we are dealing with extensive 
areas, such as the surface of a lake, it is 
necessary to remember that a level surface 
is bound to approximate to the shape of 
the earth and is therefore a portion of a 
spherical surface. 

There is a popular saying that " water FIG. 3. 

finds its own level." All liquids have this 
property, which amounts only to a tendency to flow in the 
containing vessel until the free surfaces ajre level. 

Fig. 3 shows a U-tube containing a little liquid. The free 

surface is here broken into two owing to the shape of the 

containing vessel. Yet these two 

// portions are each level when con- 

/ / sidered separately and they are both 

/ / ^ on the same level when considered 

Lr . -^3 together. 

JS ^^T If the tube is tilted, as shown in 

^l ^^T Fig. 4, the level of the water is 

^R^tol^pr maintained although the vessel is 

now in a different position. 

FIG. 4. Liquids which wet the surface of 

a containing vessel show a tendency 

to creep up the side of the vessel. This results in the liquid 
being not quite level near the edge of the vessel. 

Fig. 5 shows a few cases. It will be seen that if the vessel 
is large (as at A) the surface of the liquid is level except 
near the edge. As the surface becomes smaller, however, the 
effect is greater, 
until in very nar- 
row tubes (as C) a 
liquid will appear 
to have a surface 
which is wholly 

When making 
measurements of 



FIG. 6. 

columns of liquids the measurement is 
always made from the lowest part of the 
curved surface in the case of liquids which 
wet the vessel. 

Liquids which do not wet the vessel, such 
as mercury, have a curved surface which 
is convex (as D in Fig. 5). 

In such cases measurements are made 
from the upper part of the curve. 

Another modification of this phenomenon is observed 
when a tube (open at both ends) is placed in a vessel of 
liquid as shown in Fig. 6. The liquid creeps up the tube, 
the distance depending upon the diameter of the tube and 
the nature of the liquid. 

Experiment 8. 

By means of a few pieces of glass tube of different bores, 
observe the phenomena just described. Use a number of 
liquids, such as mercury, water, and various solutions. 

Floating Bodies. If a piece of wood is placed in a vessel 
of water it ultimately comes to rest with a portion of its 
volume immersed and the remainder above the water level. 
It is then said to be floating. Now since the immersed 
portion has, so to speak, made a hole in the water, the water 
which originally occupied this space has necessarily been 
obliged to find accommodation elsewhere. It has, of course, 
caused a raising of the level of the water. 

Fig. 7 shows a sphere of wood floating in a vessel of water. 
The shaded portion of the wood indicates the amount which 

is immersed and the shaded 

portion of the water shows the 
amount of displaced water, the 
water having risen from A to B. 

The volumes which these por- 
tions represent must of necessity 
be equal. 

Now it is reasonable to sup- 
pose that if we push a quantity 

FIG. 7. 



FIG. 8. 

of water out of the position it 
naturally occupies, it will make 
an effort to return. Hence the 
weight of the displaced water 
forms a force which tends to 
push the wood out of the 
water. Since the wood is float- 
ing at rest it follows that : 

Weight of wood = weight of 
displaced water. 

If a sphere of cork of the same size as that of the wood 
were placed in water, it would float with less of its volume 
immersed, for, since cork is specifically lighter than wood, 
a smaller volume of water is required to yield a force equal 
to the weight of the cork (see Fig. 8). 

If a sphere of iron were placed in water it would pass into 
the water until the whole of its volume was immersed, and 
even then the weight of the displaced water would be less 
than the weight of the iron ; consequently the iron would 
not be supported but would sink to the bottom of the vessel. 
Only bodies whose specific gravity is less 
than unity will float in water. Generally we 
may say, that bodies will float in a liquid 
provided the specific gravity of the body is 
less than that of the liquid. 

Experiment 9. 

The Hydrometer. Take a piece of glass 
tubing about 8 inches long and seal up one 
end. Place in it a few lead shot and above 
these a paper scale. (A roll of ordinary 
squared paper makes a suitable scale.) 

If such a tube be placed in a vessel of 
water as shown in Fig. 9 the tube will float 
according to the law already given. That is, 
the immersed portion will displace a volume 
of water equal in weight to the weight of the 
FIG. 9. tube, etc, 



It is readily seen that the position of the level of the liquid 
as shown by the scale enclosed in the tube depends upon the 
specific gravity of the liquid. The denser the liquid the less 
will the tube need to sink in the liquid. Such a tube is called 
a hydrometer. 

By varying the quantity of shot, a number of hydrometers 
may be constructed to suit liquids ranging from very low 
to very high specific gravities. 

If the records of these instruments are compared with the 
specific gravity of the liquids obtained by means of a specific 
gravity bottle it is possible to replace the arbitrary scale of 
squared paper by a scale directly recording specific gravities. 

Ex. 8. The following table shows the arbitrary readings 
of a glass tube hydrometer in liquids of known specific gravity. 
Prepare a scale of specific gravities for the hydrometer : 
Specific gravity . . 1* 1*15 1*21. 

T T OK" K "I " K Q " 

Reading . . .36 61 o o . 

First plot a graph of these results. 


1-0 1-05 1-1 1-15 

FIG. 10. 



This is shown in Fig. 10, and it will be seen that in this 
case it is a straight line. From the graph it is possible to 
read the specific gravity corresponding to any scale readings. 
Thus we may compile a table as follows ; 



Specific gravity. 



Corresponding Scale 

3 -5* 



If a scale is now prepared on which the above specific 
gravities replace the positions originally occupied by the 
corresponding scale readings we have a hydrometer which 
will record specific gravities directly. 

It may be mentioned that a hydrometer may be made 
more sensitive by having a bulb blown at or near the bottom. 
Of course, the more sensitive we 
make the instrument, the smaller 
will be the range of its readings. 

Determination of the Density 
of Irregular Solids. We are now 

in a position to determine the 
density of solids of irregular 
form. Consider a piece of iron 
whose volume is, say, 10 c.c. If 
this be dropped into water it 
will displace 10 c.c. of water 
which will weigh 10 grammes. 
As we have seen, this displaced 
water will exert an upward thrust 
of 10 grammes on the iron. As 
the iron itself weighs more than 
this the force will be insufficient 
to float the iron. Nevertheless, 
if the iron were weighed while 
it was suspended in water it 
would be found to weigh 10 
grms. less than its normal weight. 
We can make use of this fact 
to obtain the volume of an FIG. 11, 


irregular body. Fig. 11 shows a simple arrangement for 
obtaining the weight of a body while suspended in water. 
A light wooden bridge spans the pan of an ordinary laboratory 
balance. On the bridge rests a vessel of water, and the body, 
suspended by a fine thread from the hook of the knife edge, 
is wholly immersed in the water. Its weight is now obtained 
in the usual way. 

Ex. 9. . A piece of lead weighs 257*928 grms. When 
suspe'nded in water it weighs 235*295 grms. Find the density 
of lead. 

Weight of lead 257 '928 grms. 

Weight of lead when suspended in water . 235*295 ,, 

Upward thrust due to displaced water . . 22*633 ,, 

As this is the weight of the displaced water and as 1 grm. 
of water has a volume of 1 c.c. it follows that the volume of 
the displaced water (and therefore of the lead) is 22*633 c.c. 

Weight 257*928 grins. 
Now Density = ==-= 

Volume 22*633 c.c. 

= 11*38 grms. per c.c. 
Experiment 10. 

Find the density of a number of irregular solids, avoiding 
those which would dissolve in water. 

Determination of the Density of a Powder. Powders and 
granulated solids are best dealt with by means of the specific 
gravity bottle. An example will make the method clear. 

Ex. 10. Find the density of sand, employing the specific 
gravity bottle used in Example 6. 

Weight of sand taken .... 3'148 grms. 

This was placed in the bottle, which was then filled with 
water, care being taken to remove air bubbles, and the whole 

The weight was 45*523 grms, 


Using the data of Example 6. 

Weight of bottle and water . . . 43 '728 grms. 

Weight of sand . . 3 '148 

Adding we get 46 '876 

The actual combined weight was . . 45*523 

Difference . . . 1*353 

This difference is due to the fact that the sand occupies 
space and therefore the bottle holds less water. 

The weight of water displaced = 1 *353 grms. 

The volume of this water (and therefore of the sand) is 
1*353 c.c. 

Density =^r 


_3*148 grms. 
1*353 c.c. 
=2*32 grms. per c.c. 

Determination of the Density of a Soluble Solid. Solids 
which are soluble in water have applied to them one of the 
above methods, with the exception that a liquid in which the 
solid will not clissolve is substituted for water. 

Ex. 11. A lump of salt weighs 18*412 grms. When sus- 
pended in alcohol of specific gravity 0*71 it weighs 14*108 
grms. Find the density of the salt. 

Weight of salt . 18*412 grms. 

Weight of salt suspended in alcohol . . 14*108 ,, 

Upward thrust due to displaced alcohol . 4*304 ,, 
Weight of displaced alcohol . . . 4*304 

Weight in grammes. 
Volume m c.c. = ^ 

Specific gravity. 


= 6*06 c.c. 



This being the volume of displaced alcohol is likewise; the 
volume of the salt. 

Now density =^^ T 


_ 18 -412 grms. 
6-06 e.c, 

=3 '04 grms. per c.c. 

Hare's Apparatus. Let two glass tubes be supported 
vertically with their lower ends respectively in two beakers 
containing different liquids as shown in Fig. 12, A and B. 

Let the upper ends of the tubes be joined by a T piece 
(C) fitted with a tap. If the tap be opened to the air the 
liquid in the tubes will stand at the same height as that in 

its beaker (except for the creeping 
up effect called " capillary attrac- 
tion," and this will be negligible 
unless the tubes are very narrow). 
Suppose air is withdrawn from 
the tubes by suction through C 
and the tap closed. A certain 
amount of each liquid will be 
drawn into the tubes, and since 
the remaining air is free to pass 
from one tube to the other it 
follows that the two supported 
columns of liquid must balance. 

If therefore the vertical height 
of each column above the level 
of the liquid in the beaker be 
measured it is possible to compare 
the density of the two liquids. 

In Fig. 12 let the tube A dip 
into pure water and the tube B 
into a solution of salt. Since 
the latter solution is specifically 
FIG. 12. heavier it follows that a shorter 



column of it will be supported. It is easy to see that the 


specific gravity of the salt solution is given by -=- 

This is quite obvious if the two tubes are of exactly the 
same bore, but experiment shows that it is also true when the 
tubes are of different bore. It is not even necessary for the 
tubes to be of uniform bore. An explanation of this will 
be given in the next chapter. 

Experiment 11. 

Make a Hare's apparatus using two tubes of the same 
bore and find the specific gravity of salt water. 

Repeat the experiment, using tubes of different bore, but 
the same salt water. Compare results. 

In experiments of this nature the student should not be 
satisfied with a result obtained from a single reading. Several 
readings should be taken with columns of different heights 
and a mean value secured. 

The Syphon. Fig. 13 illustrates two syphons, which 
consist of bent pieces of tube each having one limb longer 
than the other. If such a tube be filled with a liquid and 
the short limb placed in a vessel of the liquid, the columns, 
being unequal in length, will not balance. The longer one, 
falling, wifi pull the shorter one after it, and thus liquid will 


be drawn out of the vessel. It will be shown subsequently 
that it is the vertical height of such a column which is effective, 
and thus the two syphons illustrated would be of equal 

The difference in the lengths of a and 6 represents a column 
of liquid, the weight of which gives the effective suction. 

Experiment 12. 

Make a number of syphons and compare the amount of 
water which is taken from a vessel per minute with the value 
of (b a) in Fig. 12. 

Exercises 3. 

1. 25 cubic centimetres of copper sulphate solution were 
poured into a weighed evaporating basin. Determine the 
density of the solution from the following results : weight 
of basin 55*152 grammes, weight of basin plus 25 cubic centi- 
metres of copper sulphate solution, 83 '805 grammes. 

2. A steel ball 10 millimetres in diameter was placed in a 
burette containing water and its volume was found to be 
0*35 cubic centimetres. Is this correct ? The weight of the 
ball was 4 '47 grammes. Determine the true density of the 

3. A specific gravity bottle weighs 15*67 grammes and 
contains 66 cubic centimetres of water when full. Some 
vaseline was warmed until it flowed readily and then poured 
into the specific gravity bottle until it was full. The weight 
of the bottle was then found to be 72*372 grammes. Deter- 
mine the density of the vaseline. 

4. In determining the density of mercury by the "U" 
tube method the following readings were obtained, measure- 
ments being taken from the same level in each case. Height 
of water, 15*5 centimetres. Height of mercury, 1*14 centi- 
metres. Determine the density of mercury. 

5. An experiment was carried out to determine the density 
of mercury and the following results were obtained : Weight 
of beaker 29 '21 grammes. Volume of mercury weighed, 19 


cubic centimetres. Weight of the beaker and the 19 c.c. 
of mercury 285 '2 grammes. Determine the density of the 
mercury from these results and explain how you would 
carry out a similar experiment. 

6. Sketch and describe Hare's apparatus and explain how 
you would determine the density of mercury with this ap- 
paratus. The following results were obtained with the 
apparatus : Height of water 24 '5 centimetres, height of 
mercury 1/8 centimetres. Calculate the density of mercury 
from these results. 

7. A piece of iron weighed 26 - 734 grammes in air and 23 '319 
grammes when weighed in water. Determine its specific 

8. The specific gravity of a piece of wood was required and 
the following experimental results were obtained : 

Weight of the body in air, 7 '735 grammes. 

Weight of the sinker in air, 26 '735 grammes. 

Weight of body plus the sinker in water, 16*05 grammes. 

Weight of the sinker in water, 23 '32 grammes. 

Determine the specific gravity of the material and explain 
with the aid of sketches how you would carry out a similar 

9. Hare's apparatus was used to compare the density of 
benzine with turpentine, and in carrying out the experiment 
the following readings were noted : Height of benzine llf", 
height of turpentine 9fV'. Compare the density of benzine 
with that of turpentine and the converse. 

A further experiment was carried out with benzine and water 
and the following readings were taken : Height of benzine 
11 "6", height of water 10 '2". Determine the density of the 

10. The following results were obtained after carrying out 
the experiment in question 1. Weight of the beaker holding 
copper sulphate 41 '1 grammes. Weight of the beaker plus 
copper sulphate solution 155 '13 grammes. Volume of the 
weighed copper sulphate solution 100 cubic centimetres. 
Determine the density. 


At the completion of the test 100 cubic centimetres of 
water were added to the copper sulphate solution, and the 
weight of the beaker and solution, plus added water, was now 
255*075 grammes. Determine the new density. Is the 
density the mean of the density of water and that of the 
copper sulphate solution ? 

11. An experiment was carried out to determine the 
specific gravity of a wooden stopper and the following results 
were obtained : 

Weight of stopper and sinker in air . . 21 '82 grammes. 

Weight of stopper and sinker in water . 10 '58 ,, 

Weight of the sinker in air . . . 13 '85 

Weight of the sinker in water . . . 12 '55 

Determine the specific gravity of the stopper. 

12. A test tube was taken and a small quantity of lead 
shot placed in it. A piece of squared paper was then placed 
in the test tube and the test tube was placed in various liquids 
and the height of flotation noted in each case. The results 
were as follows : 

Scale reading . . 0'313 0'9 1'5 T55 

Specific gravity . . 0'92 1-21 A 1*358 

Plot a graph of specific gravities and scale readings and 
determine the specific gravity of "A." 

13. An experiment was carried out to determine the influ- 
ence of the bore of the tubes in Hare's apparatus and the 
results are as stated : 

With tubes of the same bore 

Height of water in the tube . . . .5*5* 
Height of alcohol . . < . . . G'9" 

With tubes of different bores 

Height of the water ..... 5 '15" 
Height of the alcohol . . . 7 '37* 

Determine the specific gravity in each case ; does the 
bore of the tube affect the result in this experiment ? 


14. An experiment was carried out to determine the effect 
of altering the bore of the tubes on the quantity of water 
delivered by various syphons. (See Fig. 13.) 

First Experiment 
Bore of the tube, 2 millimetres. 
Difference of vertical heights, 0'66". 
Outflow in cubic centimetres per minute =75. 

Second Experiment 
Bore of tube, 4 '5 mm. 
Difference of the vertical heights 1 '5". 
Outflow in c.c. per min. = 142. 

Third Experiment 
Bore of tube, 4 '5 mm. 
Difference of vertical heights, 1 '15". 
Outflow in c.c. per min. =412. 

Carefully examine these results and state what deductions 
can be made. 

15. An experiment was carried out to determine the specific 
gravity of glass and the weight of the glass in air was 2*85 
grammes. The weight of the glass in water was 1 '687 grammes. 
Determine the specific gravity of the glass. 



Fluid Pressure. If we consider a cylindrical vessel of 
diameter d, similar to that shewn in Fig. 14, containing a 
liquid to a depth of h, it is obvious that the volume of liquid 

is ~rd 2 h, and if the density of the liquid is unity (as in the 

case of water, which weighs 1 grm. per cubic centimetre) 
this expression gives the weight of the liquid. 

Now this weight is clearly supported by the base of the 

vessel, which has an area of d 2 ; hence the pressure of the 
vessel is h units per unit area. 

Jb'io. 14. 

Now consider a vessel of the shape shewn in Fig. 15. It 
is clear that the base of the vessel supports a cylinder of 
water of the dimensions shown by the dotted lines, that is, 
the pressure on the bottom is the same as that in the case 


just considered. The additional water is in this case sup- 
ported by the sides. 

Consider a point P which lias a depth of x units of water 
above it. If a small hole could be pierced in the walls of 
the vessel at this point the water would issue normally to the 
wall of the vessel with a force proportional to the depth x. 

We may say therefore that in this case at least the oblique 
sides of the vessel are subjected to a pressure which is directly 
proportional to the depth of water at the point considered 
and acts normally, that is, at right angles, to the surface of 
the vessel at the point. 

Lastly, consider a vessel of the shape shown in Fig. 16. 
Here a concentric circle in the base sup- 
ports a complete cylinder of water and 
the pressure on this is obviously the same 
as in the previous examples. The annular 
space round this circle has above it water 
varying in depth from o to h. Consider 
a given point. Here the pressure is 
again normal, that is, perpendicular to 
the surface. Furthermore, it is, as in 
the last example, proportional to the 
depth of w r ater at this point. 

Now it is an established principle in 
mechanics that every force is met by a FIG. 1C. 

reaction equal in magnitude and opposite 
in direction. Hence at a given point the wall of the vessel 
must be pressing downwards with a force equal to the pressure 
of the water upwards. Hence if the downward pressure at 
this point is X units, this added to the column of water 
of y units immediately below it, makes a total pressure on 
the bottom of the vessel of X-\-y units, which is equal to h. 

It appears therefore that the pressure on the bottom of 
the vessel is everywhere the same, namely, h units per unit 
area. It obviously must be so, for if it were not, water 
would flow from the region of high pressure to one of lower 
pressure, until equality of pressure was established. 

We have seen that the total pressure on the base of the 
vessel in Fig. 16 is the same as that represented in Fig. 14, 


yet the weight of water in the conical vessel is less than that 
in the cylindrical vessel. This is sometimes a little confusing 
to a student, but it is similar in principle to the case of a 
common lever. Here a small force applied at one end becomes 
a large force at the other, but the work done at both ends 
is equal. The question of work has, so far, not entered into 
our consideration of fluid pressure. 

Properties of Gases. A gas is another form of fluid, but 
a gas can not only flow ; it has also the special property of 
filling its containing vessel, irrespective of the quantity of 
gas or the size of the vessel ; that is, there is no level surface 
terminating a quantity of gas such as we found in the case of a 

Now if we put a pint of gas into a quart bottle the latter 
is only " full " in the sense that the gas is equally distributed 
all over the bottle. There is no space which contains more 
or less gas per unit volume than any other space. There is, 
however, nothing to prevent another pint of gas being put 
into the same bottle and a further pint after that. Thus a 
bicycle tyre is capable of holding a very variable quantity of 

Experience shows, however, that the more air one pumps 
into a bicycle tyre the " harder " it gets : in other words, 
the higher the pressure of the air rises. 

Another point which experience has established is that the 
air in a bicycle tyre will escape from a hole made in any part 
of the tyre. For a liquid to escape from a vessel it is neces- 
sary for the hole to be below the level of the liquid ; but a 
gas exerts its pressure equally in all directions, hence it 
will pass through a hole in the top or side of a vessel as readily 
as through a hole in the bottom. 

Atmospheric Pressure. Gases, like other forms of matter, 
possess weight. It is true that their density is comparatively 
small, but it is by no means negligible. Air, for example, 
under normal atmospheric conditions, has a density of about 
0'08 Ibs. per cubic foot. In other words a cube of edge 2 ft. 
4 ins, contains about 1 Ib, of air. 


Now just as a liquid exerts a pressure on the bottom of its 
containing vessel, so we may expect the air at the earth's 
surface to be affected by the weight of the air above it. 

The case is not, however, quite so simple, for the air in the 
lower regions, being subject to the pressure due to the weight 
of the air above it, is compressed, and is therefore much 
denser than the air in higher regions. Liquids, on the other 
hand, are almost incompressible. 

The Barometer. In considering Fig. 14 we saw that the 
total pressure on the base of the vessel was equal to the 
weight of the water. As a matter of fact, to this amount 
must be added the pressure of the air on the surface of the 
water. It is clear, therefore, that the reaction of the force 
on the base of any vessel containing a liquid is acting vertic- 
ally upwards and is equal to the pressure due to the liquid 
plus the pressure of the air. 

Now if we could remove the atmospheric pressure from a 
portion of the liquid surface, the liquid immediately below 
the surface should be forced up. Such conditions can be 
partially obtained by placing a tube in a basin of water and 
sucking some of the air out of the tube. The water at once 
passes up the tube under the action of the force. 

If we had a very long tube it is obvious that we might 
reach a limit where the pressure due to the column of water 
would be equal to the pressure of the air. No amount of 
suction could draw water up the tube beyond this limit. 

This principle may be utilised to measure the pressure of 
the air. In order to avoid the use of very long tubes it is 
usual to employ a liquid of a density much higher than that 
of water : mercury, for example. 

Experiment 13. 

Obtain a piece of barometer tube about 33 ins. long. This 
tube has stout walls to give strength. The bore depends 
upon the quantity of mercury available, but for experimental 
purposes a bore of 3 or 4 millimetres is sufficient. One end 
of this tube should be sealed in a blowpipe flame. 

This tube has now to be filled with mercury and certain 
precautions should be taken whenever mercury is used. 


Precautions in the Use of Mercury. Mercury is the only 
metal which is liquid at the ordinary temperature of the air. 
It readily amalgamates with certain other metals and should, 
therefore, be kept out of contact with other metals. In this 
connection it is a source of danger in a laboratory sink, as 
it is liable to amalgamate with (and destroy) the lead drainage 

As mercury is very costly it is desirable to support any 
apparatus containing it in a deep wooden tray made for the 
purpose. If any mercury is spilled, it is then caught in the 

Mercury does not " wet " most surfaces, and consequently 
when dropped on to a flat surface, gathers itself up into numer- 
ous little globules which are not readily collected. 

If in spite of precautions mercury does get into a sink, 
it may be removed by means of a small piece of zinc which 
has been dipped in dilute sulphuric acid. The mercury 
readily amalgamates with the zinc and is thus removed. If 
sufficient mercury is involved it may be obtained by dissolv- 
ing the zinc in dilute sulphuric acid, after which the mercury 
may be washed with water and dried. 

If mercury becomes mixed with moisture or dirt it may be 
cleansed by allowing it to pass through a funnel fitted with 
a filter paper in the apex of which a small hole has been 
pierced. The dirt adheres to the paper and the clean mercury 
passes through. 

When a long tube has to be filled with mercury, the tube 
should if possible be supported in a slanting position, and in 
any case the mercury should be allowed to pass in very slowly 
and in small quantities at a time. Mercury has so high a 
density that any considerable quantity of it falling vertically 
down a long tube strikes the end with such force that there 
is a danger of the tube breaking. 

Try to avoid air bubbles in a mercury column. With 
patience and a little gentle tapping they can generally be 

Returning to our barometer tube, when the tube is com- 
pletely filled with mercury, the open end should be temporarily 
closed (this is generally done by the thumb of the right hand), 


the tube inverted and the open 
end placed below the surface of a 
little mercury contained in a 
vessel, as shown in the left-hand 
portion of Fig. 17. 

Now it is clear that no air can 
enter the barometer tube ; hence 
there can be no air pressure on 
the surface of the mercury over 
which the tube is placed ; but 
when all flowing has ceased the 
pressure must be equal at all 
points of the surface, so that there 
will remain in the barometer tube 
a column of mercury which exerts 
the same pressure as that which 
the air is exerting on the outside 

This column is generally in the 
neighbourhood of 760 mm. or 
slightly under 30 inches, and is 
shown by h in Fig. 17. It 
should be noted that the height of the mercury column is 
measured from the level of the mercury in the dish and not 
from the bottom of the tube. It is clear that the portion 
of the tube above the mercury column cannot contain any- 
thing except the vapour of mercury, which at ordinary tem- 
peratures is quite insignificant. 

If the pressure of the air changes, mercury will flow into 
or out of the tube until the mercury column again balances 
the pressure. 

Thus we speak of the pressure of the air being indicated 
by the " height of the barometer." 

If a barometer tube be inclined as shown on the right hand 
side of Fig. 17 we find that the mercury remains at the same 
level. In other words, it is the vertical height of the column 
of mercury which records the atmospheric pressure. 

Fig. 18 shows another type of mercury barometer, working 
on what is known as the " syphon " principle. Although 

FIG. 17. 



FIG. 18. 

somewhat different in form the actual principle 
is the same. The air exerts a pressure on the 
open surface of the mercury at B and supports 
a column h units high. For purpose of gradu- 
ation any point A is selected and scales proceed 
upwards and downwards from this point. The 
levels of the two surfaces of mercury are read 
on these scales and added together. Obviously 
x-\-y=h. When a permanent barometer is 
made for purposes where some degree of ac- 
curacy is required, it is necessary to boil the 
mercury in the tube to expel all air bubbles. 
This is an operation calling for skill and ex- 
perience and should not be undertaken without 
suitable supervision. 

A portable form of barometer is made in 
which the principle involved is quite different. 
A small flat vessel made of corrugated metal has the air 
extracted from its interior and is then sealed. The air 
pressure on the outside will depress its flat surfaces to a 
degree depending on the magnitude of the pressure. The 
movement is of course very small, but by a system of levers 
it is magnified into a suitable movement of a needle over a 
scale which may be graduated in any desired units. This 
instrument is called an " Aneroid Barometer." 

Ex. 12. The normal height of the barometer is 760 mm. 
At sea level it seldom sinks below 725 or rises above 785 mm. 
Express these pressures in Ibs. per sq. in. 

Consider a tube whose sectional area is 1 sq. cm. 

The volume of mercury in the three columns is respectively 
76, 72-5, and 78-5 c.c. 

Mercury has a density of 13*58 grms. per c.c. The weight 
of the column is therefore respectively : 

76 x 13 -58 = 1,032 grms. 
72-5x13-58= 984 
78-5x13-58 = 1,066 
These weights represent the pressure per sq. cm. 


Now 1 kilogram per sq. cm. is equivalent to 14*22 Ibs. 
per sq. inch. The pressures given above are, therefore, 
equivalent to : 


- X 14-22 = 14-67 Ibs. per sq. ins. 



X 14-22 = 14-0 
X 14-22 = 15-17 

Ex. 13. What would be the normal height of a barometer 
in which water was used instead of mercury ? 

Since mercury is 13 '58 times as dense as water it follows 
that a column of water equal in pressure to that of mercury 
must be 760xl3'58 mm. or 33J ft. 

It may be mentioned here that no suction pump can 
" lift " water to a height greater than this. 

Relation between Volume and Pressure of a Gas. We have 
already seen that when the pressure of a gas is increased (as 
in pumping up a bicycle tyre) its volume is diminished. It 
remains to find the relationship which exists between the 
pressure and volume. 

Most operators of a bicycle pump will have noticed that 
when air is rapidly compressed it becomes hot, and since heat is 
likely to affect the volume of a gas, it is desirable that we should 
investigate changes one at a time. In the following experi- 
ment, therefore, steps must be taken to secure that the 
temperature remains constant. 

Experiment 14. 

Find the relation between the pressure and volume of a 
given mass of gas when the temperature remains constant. 

Fig. 19 shows a suitable form of apparatus. It consists 
of a piece of stout glass tube bent in the form of a U, of which 
one limb is much longer than the other. The short limb is 



A little mercury is poured into the tube 
through the open end, and falling into the 
bend of the U imprisons a quantity of air in 
the short limb. 

As more mercury is poured into the tube 
this air is compressed into a piece of tube of 
length I while the pressure exerted upon it is 
the atmospheric pressure plus a column of 
mercury of height h. 

It is convenient to record pressure in terms 
of the length of a column of mercury instead 
of Ibs. per sq. in. ; a transformation of units 
can readily be made when necessary. 

Owing to the method of applying the 
pressure in this experiment the process is 
necessarily a slow one, and the very small 
quantity of heat which is produced by the 
small pressures involved has ample opportunity of passing to 
the surrounding air. It is not, therefore, necessary in this 
case to employ any special means of maintaining the tem- 
perature constant. 

A number of readings should be made, commencing if 
possible with one in which the pressure is below the atmo- 
spheric pressure. This can be obtained by having the mercury 
level in the open limb lower than that in the closed limb. The 
height h would then be called a " negative head." 
The following is a typical set of results : 

Reading of barometer, 765 mm. 

FIG. 19. 

Length I in mm. 






Length h in mm. 






Now the length / may be regarded as proportional to the 
volume of the air. To the value of h the barometric reading 
must be added, and this will give the total pressure exerted 
on the volume of air, expressed in millimetres of mercury. 
Calling the volume v and the pressure p we have ; 














The student should use these results (or preferably those 
of his own experiment) to plot two graphs (i) to show the 
relation between p and v, and (ii) to show the relation between 

v and - 

He should also find the value of the product of p and v 
in each case. The best experimental work goes to show that 
the product of p and v is a constant. Another way of ex- 
pressing this is to say that the volume varies inversely as 
the pressure. This law of Nature was discovered by Boyle 
in 1662, and its enunciation is known as Boyle's Law. 

Boyle's Law. If the temperature remain constant the 
volume of a given mass of gas varies inversely as the pressure. 

Ex. 14. A quantity of gas is measured under a pressure of 
834 mm. of mercury and found to have a volume of 15 '7 cc. 
What volume would it occupy at a pressure of 760 mm. and 
at what pressure would its volume be 12 c.c. ? 

Nowv =15 '7 when p = 834. 
/. pv =834 x 15-7 -13,100. 

But this product is a constant. 

Hence Vx 760 = 13,100. 
Where V is the required volume. 


Also P x 12 = 13,100. 
Where P is the pressure corresponding to a volume of 12 c.c. 


Pumps. A pump is a mechanical device for transporting 
a fluid from one place to another. Pumps may conveniently 
be divided into two classes, rotary and reciprocating. In 
the former, all the moving parts rotate, while in the latter 
some at least of the moving parts " reciprocate " ; that is, 
move to and fro along a straight line. 

Valves are always associated with reciprocating pumps, 
a valve being a passage through which the fluid can flow only 
in one direction. 

A common type of pump for dealing with air is the bicycle 
pump which is illustrated in Fig. 20. It consists of a cylinder 
with a plunger or piston. The latter consists of a disc of 
thin leather formed into the shape of a cup, the edges of which 
are turned towards the delivery end of the pump. 

On the downward stroke M 

the edges of this piston are 

pressed against the sides of 

the cylinder so that no air can 

pass. Thus the air is com- JT IG> 20. 

pressed into the lower part of 

the pump until the pressure is reached which is necessary to 

open the valve of the tyre. (This pressure of course depends 

upon the pressure of the air already in the tyre.) 

On the upward stroke, since no air can enter the pump 
from the delivery end (owing to the valve in the tyre) the 
pressure of the little air remaining in the pump soon falls, 
and is less than atmospheric pressure (that is, there is a 
partial vacuum). In consequence air pushes past the skirts 
of the piston, since these are not now being pushed against 
the cylinder walls, from the upper part of the pump which 
is open to the air. Thus one gets a cylinder full of air ready 
for the next compression stroke. 

It is easy to see that as the pressure in the tyre gets greater, 
more and more of the compression stroke will have to be 
performed before the pressure rises sufficiently high to open 
the tyre valve, and the higher will be the pressure of the 
residual air left in the base of the pump. 

A Common Air Pump. A bicycle pump is used to pump 



air into a receptacle, and may be called a 
" compressor." Fig. 21 shows a type of air 
pump which is used to extract air from a vessel 
and may be called a " decompressor." 

Again it consists of a cylinder and piston. 
There is one valve at the end of the cylinder A 
and another in the piston B. These valves 
generally consist of a small hole over which a 
piece of oiled silk is stretched, this being placed w 
on the side away from the vessel to be exhausted. 

On the outward stroke, valve A opens and B 
closes. A partial vacuum is created in the 
cylinder and air flows from the vessel to the 
cylinder. On the return stroke A closes and 
B opens, and this air passes out into the 

It is clear that as the pressure in the vessel 
falls, a higher and higher degree of vacuum has 
to be created in the cylinder before valve A will 
open, and thus the pump becomes less and less 
effective as the operation proceeds. 

A Lift Pump for Water. Fig. 22 shows a common form of 
water pump. There are two valves, both of 
which can only open upwards. They are often 
flap valves which operate after the manner of 
a door and normally remain closed by the 
operation of their own weight. One valve is 

^ aa=e \ placed at the bottom of the cylinder and the 
other is in the piston. 

The diagram shows the piston on its down 
stroke. This causes the bottom valve to close 
and the water in the cylinder (being practically 
incompressible) opens the valve in the piston 
and flows through and ultimately passes out 
through the spout. 

On the up stroke the valve in the piston is 
closed and as soon as a partial vacuum is 
FIG. 22. created in the cylinder the lower valve opens 



and water flows up (or is "lifted"). This 
flow is due entirely to the pressure of the 
air and hence it is impossible to " lift " 
water to a height greater than 33^ feet (the 
height of a " water barometer "). 

Since such a pump could never produce a 
perfect vacuum the limit of lift in practice 
is very much less than this amount. 

A Force Pump for Water. When water 
has to be raised from a deep well or a 
mine, it is necessary to have a force pump. 
Such a pump is shown in Fig. 23, and it 
will be observed that there is no valve in 
the piston. The valves are fitted, one at 

A opening inwards and one at B opening outwards. The 

pump must be placed sufficiently near the water for the 

suction stroke to lift it. 

The diagram shows the piston on the up stroke. A is 

open and B closed and water is flowing into the cylinder. 

When the piston reaches the top the cylinder is full of water. 
On the downward stroke the compressiqn causes valve A 

to close and valve B to open. Hence the water flows through 

B and up the outlet pipe, which ^ 

may rise to any height whatever, 

provided the power supplied to 

the pump is sufficient to force 

the water out of the cylinder. 

A Power-Driven Air Compressor. 

Compressed air is used for so 
many purposes in modern en- 
gineering practice that it is 
necessary to have power- driven 
air compressors. A diagrammatic 
illustration of such a pump is 
shown in Fig. 24. 

It is fitted with two valves 
of mushroom shape which lift 

FIG. 24. 


under air pressure against the action of a spring which 
normally keeps them closed. The piston is shown on the 
down stroke and air is being forced out of the cylinder into 
a receptacle. On the return stroke the outlet valve closes 
and the inlet valve opens, enabling air to be drawn into 
the cylinder ready for compression on the following 

As these pumps often work at high pressures and speeds, 
the compressed air becomes very hot and is liable to give 
trouble. To meet this difficulty it is a common practice 
to place round the cylinder a jacket through which a stream 
of cold water is passed in the direction indicated by the 
arrows in the diagram. 

Sometimes air is compressed in a pump of this nature, 
then cooled and passed into another pump which compresses 
it to a still higher pressure. This is called a " two stage 

Diffusion. Vessels made of unglazed earthenware and 
called " porous pots " are used in connection with the making 
up of electric cells. If one of these (preferably a small 
cylindrical one) is taken and fitted with a cork through which 
a glass tube passes it will be found that by blowing down the 
tube the cylinder appears to be " air tight." Indeed, it is 
possible to maintain a considerable pressure in such a vessel 
for some time. 

If, however, such a vessel is filled with water it . will be 
found that in an hour or two damp patches appear on the 
outer surface, showing that the walls of the vessel possess 
small pores through which water can pass, although the 
process is a very slow one. 

If a porous pot, fitted with a cork and tube, be supported 
in an inverted position with the open end of the tube below 
the surface of a little water (preferably containing colouring 
matter) as shown in Fig. 25, the pressure of the gas within 
the porous pot cannot change without a visible movement 
in the water. That is, if the pressure rises, bubbles will pass 
from the tube through the water, while a fall of pressure will 
cause the water to pass up the tube. 



FIG. 25. 

If a bell jar be placed over the pot as shown 
in the figure, we have air within and without 
the porous pot. Now pass hydrogen or other 
light gas (coal gas will answer) into the bell 
jar. Immediately bubbles begin to pass through 
the water, which seems to indicate that some of 
the hydrogen has found its way into the porous 

There is no question here of gas being forced 
through the walls of the vessel by pressure for 
the whole process is conducted at atmospheric 
pressure. The phenomenon is known as diffu- 
sion and depends upon a difference of density, 
and not a difference of pressure. 

It is found that all gases can pass through 
the walls of a porous vessel at a rate inversely 
proportional to the square root of the density 
of the gas. Hence, in the case just given, 
hydrogen, having a density approximate^ one-fourteenth 
that of air, entered the porous pot at about A/14 (=3|) the 
rate at which the air passed out. This accounts for the 
escape of some of the mixture in the form of bubbles. 

After a time the porous pot will contain air mixed with a 
considerable quantity of hydrogen, and if the bell jar is re- 
moved this mixture will be less dense than the air outside, 
and consequently gas should pass out more quickly than air 
enters. This is indicated by the coloured water rising up 
the tube. 

The explanation of the phenomenon of diffusion depends 
upon the molecular theory of gas and need not concern the 
student at the present stage of his work. The phenomenon 
itself is, however, too important to be neglected. 

Experiment 15. 

Fit up the apparatus shown in Fig. 25 and carry out the 
experiment just described. By replacing the gas jar by 
a jacket with its open end upwards, it is possible to try the 
effect of surrounding the porous pot with a heavy gas, such as 
carbon dioxide. 



Exercises 4. 

1. A cylindrical tank is 2' 3" in diameter and 2' 6" deep, 
and is full of water. Determine the pressure per square 
inch on the base. 

2. A rectangular tank is 4' 6" long and 2' wide, and 1J" 
deep. Determine the pressure per sq. in. on the base. 

3. The absolute pressure in a condenser may be measured 
by means of a glass rod standing vertically in a bowl of 
mercury. When the upper end of the tube is connected 
with the condenser the mercury rises up the tube. If it 
rose 21" on a day when the barometer stood at 758 mm., 
find the absolute pressure in the condenser. 

4. The barometric pressure is 29 '5" of mercury. There is 
a vacuum in the condenser of 25" of mercury. Find the 
atmospheric pressure and absolute pressure in the condenser 
in Ibs. per sq. in. 

5. Water is contained in a tank at the top of a building and 
the surface of the water is 30' above ground level. Determine 
the height of a mercury column just to balance this head of 
water at ground level. 

6. In a test on a small blower a " U " tube containing water 
was used to measure the difference of pressure between the 
air in and outside a large box. The following readings were 
taken : "395, '43 1 , '464 inches of water. If the barometer 
was at 760 mm. determine the excess pressure inside the 

7. The following readings of pressure and volume were 
obtained with a Boyle's Law apparatus. From these results 
determine the average value of the constant and plot a graph 
of pressures and volumes. 

Pressure cms. 
of mercury. 



114-5 105-8 





in cc. 



15-7 17 





8. By means of the " U " tube determine the pressure of 
the coal gas supply from the main, in inches of water. 

9. In a boiler test the force of the draught in inches of water 
was 0'38. What is the pressure in Ibs. per sq. in. ? 

10. Give a sectional drawing of a bicycle pump. Assuming 
the pump described has 100% efficiency, measure the diameter 
and stroke of such a pump and state what volume of air at 
standard temperature and pressure it can supply each stroke. 

11. Explain by the aid of sketches how any small glass 


12. Give a diagrammatic sketch of a self -filling fountain 
pen and explain its action. 

13. In using a pipette explain why the liquid flows upward 
when the air is sucked away from the top of the tube. Why 
does the liquid stop running out of the pipette when the 
finger is placed over the top ? 

14. A cylindrical tin can is filled with water and the lid 
is soldered on. A small hole is then bored in one end near 
the outside rim. The can is placed in a horizontal 
position with the hole near the ground. Will all the water 
run out of the can ? Carry out the experiment. Explain 
what must be done to get liquid out of a barrel after the 
tap has been fixed in position. 

15. A " U " tube containing oil was used to measure the 
pressure inside a box as compared with that of the outside 
air ; the air being taken from the outside air into the box 
and thence going to a gas engine. The oil in the " U " tube 
had a head of oil 0'16. 

Determine the equivalent water column in inches, if the 
specific gravity of the oil was 0*9. 

16. Take an ordinary leather " sucker." Calculate the 
maximum weight it can lift. Test this experimentally. 
Assuming that the leather is 2" dia. determine the weight 
you would expect this to lift. 

17. Take a tin can with a narrow neck such as a " Filtrate 
Oil " can, and clean it. Add a little water and boil the water 



for some 
the neck, 

time. Remove the flame and insert a cork into 
What will happen to the can ? Try this experiment - 

18. Two surface plates as used in the metal workshop are 
placed with their finished faces together. What happens 
when you try and pull them apart ? Repeat this experiment 
with two pieces of plate glass and determine the force per 
sq. in. required to separate the two pieces. 











ing pressure 
in Ibs. sq. in. 








The above table gives barometer readings and corre- 
sponding pressures in Ibs. per sq. in. Plot the graph and 
determine a constant which multiplied by the barometer 
reading will give the corresponding pressure in Ibs. per sq. in. 

20. Plot the graph of height in feet above the earth's 
surface and pressure of the air in millibars. A " millibar " 
is a pressure of 1000 dynes per square centimetre. 

Height in ft. 





Mean pressure 

in millibars 






Convert the pressures in millibars into inches of mercury. 
21. Certain boiling points for thermometer calibrations are 


Pressures in ins. of mercury. 


Temperatures in degrees F. 



Convert the readings of pressures into millimetres of 
mercury, and those of temperature into degrees Centigrade. 
Express your results graphically. 

22. Determine how many inches of mercury the following 
pressures in millibars are equal to : 

100 120 140 160 and 180. 

23. A bicycle pump has a stroke of 11 -5" and a bore of 
0'55"i What is the displacement of the pump ? 

24. A bicycle tyre has a mean dia. of 26". It is circular 
in cross-section and 1-14" diameter. How much air at 
atmospheric pressure and temperature will it hold ? 

25. 1 gramme per cubic centimetre = 62 -43 pounds per 
cubic foot. Density of mercury at C. = 13*5955 grammes 
per cubic centimetre. 

Express the following barometer readings in millimetres 
as pressure in Ibs. per sq. inch : 735 742 758 765 770 

26. In a table giving the thermal properties of water 
the following figures are given. Complete the table. 


lbs./(feet). 3 

grammes/(centimetre) . 3 


27. The following particulars are given of a single cylinder, 
single stage air compressor. Stroke in millimetres, 250 ; 
diameter of air cylinder, 240 mm. Determine how many 
cubic feet of air at atmospheric temperature and pressure 
should theoretically be delivered in one minute if the com- 
pressor makes 125 revs, per minute. 

28. The following table shows the performance of suction 
pumps at a water temperature of 40 F. with the barometer 
at 29-92 ins. 


Plot a curve of probable actual lift and vacuum in suction 
pipe in inches of mercury and state what lift you expect at 
30 " vacuum. 

Vacuum in suction pipe Theoretical lift Probable actual lift 

in ins. of mercury. in feet. in feet. 

2 2-2 1-8 

6 6-7 5-4 

10 11-3 9-0 

14 15'8 12-6 

18 20'2 16-1 

20 22-5 18-0 

24 27-0 21-5 

28 31-6 25-2 

29 32-7 26-1 

29. Give sketches illustrating any type of pump used in 
motor -car engines to keep the cooling water for the engine 
jacket in circulation. 



Periodic Motion. If a uniformly rotating steam engine is 
observed, it will be seen that the cross -head (which connects 
the piston-rod with the connecting-rod) moves to and fro within 
its guides with a velocity which is very far from uniform. 

The movement is usually too fast for measurements to be 
made while the engine is running, but data for a space-time 
graph may be obtained by rotating the crank -shaft by hand 
through intervals of 30 (commencing with the crank at right 
angles to the axis of the cylinder), and recording the corre- 
sponding position of the cross -head. The following results 
were obtained on an engine having a stroke of 135 mm. 

Crank Angle . 

30 60 




Distance of cross - 
head from initial 
position, in mm. 

31-4 51-6 

i i 






240 270 

300 330 360 




-65-3 -36-0 


The outer dead centre is reached when the crank angle is 
90 and the inner dead centre is reached with a crank angle 
of 270. It will be observed that distances measured out- 
wards from the initial position of the cross-head have been 




regarded as positive and those are taken as negative which 
have been measured inwards from the initial position. 

Fig. 26 shows a graph of these results, and deserves careful 
study. It will be observed that the slope of the graph (which 
indicates the velocity of the cross-head) is constantly changing. 
Also note that the distance moved from the initial position 
to the outer dead centre is considerably less than that from 
the initial position to the inner dead centre. 

FIG. 26. 

The graph possesses many features of irregularity. It 
will be seen that at the points marked A, B, and C on the 
graph the cross -head is in its initial position. At B the con- 
ditions are different from those of A in that the cross-head 
is travelling in the opposite direction. 

At (7, however, the cross -head is not only in its initial 
position, but all the conditions are exactly the same as those 
which obtained when the cross -head was at the point repre- 
sented by A in the graph. In other words, if the readings 
were continued, the graph would repeat itself over and over 



A graph which possesses this property is called a " periodic - 
graph " and the motion which it expresses is called " periodic 

Simple Harmonic Motion. Fig. 27 shows the space -time 
graph of another body whose movement possessed fewer 



FIG. 27. 

irregularities. The first half of the graph is symmetrical 
about the line P Q and the whole graph is symmetrical 
about the line R S except that the latter half is below the 
axis X. 

This graph is called a " sine curve " 
and any moving body whose space-time 
curve is a sine curve is said to possess 
" simple harmonic motion." 

Ex. 15. Draw any circle having a dia- 
meter about 3" long. Draw a horizontal 
diameter as shown in Fig. 28. Mark a 
point P vertically above the centre and 
suppose that this point P travels round 
the circumference of the circle with FIG. 28. 



uniform velocity. Suppose another point travels to and 
fro along the horizontal diameter in such a way that it 
is always immediately above or below the point P. Thus 
when P has rotated through an angle of 6, that is, when it 
has passed to A, the point on the diameter has travelled 
from to B. 

Now make a table showing the corresponding values of 
and the length of OB. Thus : 

Circle of 10 cms. diameter. 

Value of 6 in degrees. 

Length OB in cms. 






The point on the diameter moves to and fro with simple 
harmonic motion. Plot a space -time curve to prove this. 

The Pendulum. A thread is supported at A (Fig. 29) and 
carries a small heavy weight at B. Such an arrangement 
is called a " plumb line " and when at rest the thread will 
hang in a vertical direction. If we pull the weight to one 
side it is obliged to travel along the arc of a circle whose 
radius is equal to the length of the string. This necessitates 
the weight being raised above its initial level. 

Thus in Fig. 29 the weight in travelling from B to C is 
raised through a vertical distance equal to B D. If the 
weight is now released the earth's gravitational force will 
tend to pull it down along the arc of the circle, and will impart 


to it such a velocity that when B is 
reached the weight cannot stop, but is 
carried up an arc on the other side of 
the initial position until all its kinetic 
energy is expended. It now repeats the 
swing in the opposite direction. 

We have here a " pendulum." If 
the thread possesses certain properties 
(among them no weight), and if the sus- 
pended weight has no size, the system 
is called a " simple pendulum ; " other- 
wise it is called a " compound pendulum." 
For most practical purposes a small 
metal weight attached to a piece of 
thread or cotton may be regarded as a 
simple pendulum. 

Experiment 16. 

Suspend a 5 or 10 gramme weight by 
a length of about 40 or 50 cms. of cotton. 
The point of suspension is loest secured 
by clamping the cotton between two 
strips of wood. 

Now measure the length of the pendulum from the point 
of support to the centre of the weight. Call this length /. 

The pendulum should now be made to swing over a small 
arc and the time required for it to make, say, 20 complete 
swings noted. (A complete swing is a swing to and fro.) 

If this time is divided by the number of swings we obtain 
the " period " (expressed in seconds), called t. 

Now change the weight, using a 20 or 50 gramme weight, 
but keeping the length as nearly as possible unchanged. 
Repeat the experiment and ascertain whether the magnitude 
of the weight has any influence on the period. 

Experiment 17. 

Set a pendulum swinging and record the time taken to 
accomplish 10 swings. The pendulum will now be swinging 
over a smaller arc, owing to friction. Without touching it, 




record the time taken for a further 10 swings and repeat the 
record until the movements of the pendulum are too small to 

Is the period of a pendulum affected by the length of the 
arc over which it swings ? 

Experiment 18. 

Using lengths from 25 cms. to 100 cms. compile a table 
showing the corresponding values of the length I and the 
period t of a pendulum. 

Plot a graph showing the relation between t and I. 

Plot another graph showing the relation between t and V^L 

It can be shown mathematically that : 


where t and I have the meaning already given to them and 
g is the acceleration due to gravity. 

g has a value of 32 '2 ft. per sec. per sec. or 981 cms. per 
sec. per sec. I must be expressed in feet or centimetres, 
according to which unit is employed for the value of g. 

Torsional Pendulum. Another example 
of periodic motion similar to that of the 
pendulum just described is furnished by a 
heavy cylinder of metal suspended on a wire 
as shown in Fig. 30. 

This is called a " torsional pendulum." 
If the metal cylinder is rotated through a 
small angle (about an axis passing along 
the wire) it will cause the wire to twist. 
This torsion of the wire will exert a force 
tending to rotate the cylinder in the 
opposite direction. When the cylinder 
reaches its initial position its acquired 
velocity causes it to twist the wire in the 
opposite direction and thus a rotational 
oscillation is set up. 

r~ ~r 

FIG. 30. 


Experiment 19. 

Construct a torsional pendulum and find its period t. 
Does the weight or size of the cylinder or the length or diameter 
of the suspending wire have any influence on the value of t ? 
(Students who are interested in the mathematical or mech- 
anical side of the subject will find a fuller treatment of this 
subject in the authors' Second Course in Mathematics 
for Technical Students, Chapter viii. It may be shown that 

I-* I T - 

where 7 is the moment of inertia of the cylinder and c is 
the value of the restoring couple due to the torsion of the 
wire when the cylinder is rotated through 1 radian.) 

The Balance-Wheel. Lastly, as an example of periodic 
motion, me ntion maybe made of the " balance-wheel." 
This consists of a delicately -balanced wheel, which in rotating 
has to overcome the resistance of a spring. This resistance 
brings the wheel to rest, and the action of the spring rotates 
the wheel in the opposite direction. It " over-shoots the 
mark " so to speak, as in the previous cases, and the spring 
pulls it back again. Thus an oscillatory motion is established. 

The regularity of the periodic motions just considered 
has been applied to the purpose of controlling the movement 
of time-keepers. The common pendulum was first used for 
this purpose by Galileo during the latter part of the sixteenth 

The torsional pendulum has been used in certain forms of 
clocks during recent years. The balance-wheel is employed 
in watches and portable clocks. 

Wave Motion. If a rope is secured to some elevated posi- 
tion and the free end pulled moderately tight and shaken, 
" waves " appear to run along the rope towards the fixed 
end. The form of the waves in the rope is shown in Fig. 31. 

Now it is clear that the material of which the rope is made 
cannot be travelling along towards the fixed end, otherwise 
there would be an accumulation of material at that point. 



A little thought should make it clear that any particle in 
the rope merely moves up and down on a path similar to 
that shown as X Y in Fig. 31. Hence the wave travels along 
the length of the rope, but any particle of the rope merely 
moves to and fro along a comparatively short path at right 
angles to the direction in which the wave is moving. 

FIG. 31. 

Transverse Wave Motion. Such u wave propagation as that 
just described in a rope is called a transverse wave motion, 
because the particles of the medium conveying the w r ave 
are moving in a direction transverse (or at right angles) to 
the direction of the wave motion. 

Another example of transverse wave motion is furnished 
by the waves formed on the surface of a liquid. The sea, 
viewed from the end of a pier, often has the appearance of 
flowing rapidly in a certain direction, but if a cork is dropped 
on to the surface of the water it usually rides up and down 
only, while the waves themselves pass on. Again, wind 
forming waves on the surface of a river often gives it the 
appearance of flowing up-stream. In this case the waves 
are travelling in one direction while the water itself is trav- 
elling in the opposite direction. 

Longitudinal Wave Motion. When a train consisting of a 
large number of trucks with link couplings is travelling with 
uniform velocity the links are all in tension. If the engine- 
driver now shuts off steam and applies the brakes to his engine 


for a moment, the velocity of the engine is reduced and the 
first truck in the train bumps into the rear of the engine, 
and rebounds. This action impairs the velocity of that truck 
and the second truck bumps into the rear of the first and so 
the bumping process is earned right through the train. 

Such a motion is another form of wave motion, but since 
the motion of the particles of the medium is in the same 
direction as the motion of the wave it is called longitudinal 
wavei motion. 

If a series of little balls were suspended on threads of equal 
length at equal distances apart, they would have the appear- 
ance of the row of dots in the upper part of Fig. 32. If 
now the end ball were set swinging after the manner of a 
pendulum so that the plane of its motion was the same as 

ABC f 

FIG. 32. 

that containing the centres of the balls, it would bump into 
the second ball and set it swinging, and this would convey 
the motion to number three and so forth, very much after 
the manner of the bumping trucks in the train. 

Now suppose at a particular moment the row of balls were 
photographed instantaneously, they would present an appear- 
ance similar to that in the lower part of Fig. 32. It will be 
seen that instead of the balls being equidistant they are 
crowded together at certain points, such as B, and spread 
out at others such as C. 

A crowded group, such as B, would appear to travel through- 
out the row, just as the crest of a wave appears to travel 
over the surface of water, but actually each ball moves over 
a very short path only. For example, the ball at F has its 
position of rest at G and its next movement will be back to 
6r, and then an equal distance beyond G. 

Amplitude and Wave Length. In both forms of wave 
motion the maximum distance which a particle moves from 


its position of rest is called the "amplitude." Thus in Fig. 31 
A is the amplitude and in Fig. 32 d is the amplitude. 

Any two particles which are in similar positions and are 
about to perform the same set of motions are said to be in 
the same " phase." Thus in Fig. 31 P and Q are in the same 
phase, so also are R and S. In Fig. 32 A and C are in the 
same phase. It may be noted that in Fig. 31 the particles 
T and V are in the same relative positions, but they are not 
in the same phase because one is about to move upwards 
and the other downwards. For a similar reason A and B 
in Fig. 32 are not in the same phase. 

In either case the distance between any point and the next 
one in the same phase is called a " wave-length," and is 
denoted by the Greek letter lambda (X). Thus in Fig. 31 
the wave-length is indicated by PQ and in Fig. 32 X = A C. 

Frequency and Velocity. The number of complete waves 
which appear to pass a given point in unit time is called 
the frequency. It is usually denoted by N. If V is the 
velocity of the wave motion it is easy to see that V= JVX. 

Exercises 5. 

1. An engine has an infinitely long connecting-rod and the 
following data for a space-time graph were obtained (commen- 
cing with the crank at right angles to the axis of the cylinder). 

Crank angle 30 60 90 120 150 180 
Distance of crosshead from the initial position 

0-5 -866 1 -866 '5 

Crank angle 210 240 270 300 330 360 
Distance, etc. -'5 '866 1 -'866 '5 

Plot a space-time graph. Compare this graph with Fig. 
27, which illustrates the graph of a simple harmonic motion. 

2. An engine has a crank 1' 0" and a connecting rod 3' Q" 
long. Commencing measurements with the crank at right 
angles to the axis of the cylinder the following results were 
obtained : 


Crank angle 30 60 90 120 150 180 210 
Distance of the crosshead from the initial position 

'45 -725 -825 '725 '45 '55 

Crank angle 240 270 300 330 360 

Position, etc. 1 M75 1 '55 

Plot a space-time graph. Compare the graph with Fig. 27. 
Describe any difference you note between this graph and 
the graph in the preceding question. 

3. Draw a horizontal line O X on a piece of paper. Take 
a thin piece of wood and fix a pin at the point and 2" 
away from O fix a drawing-pin at P, carrying a piece of 
cotton, to which a weight is attached. Place the pin at 
O in the line X and revolve the wood OP, noting every 30 
where the plumb line fixed at P intersects the line OX. 
Call the distance OM . Tabulate the results as follows : 

Angle OP makes with OX 30 60 90 120 
Distance OM measured from 2" 1-732 1" 

Complete the graph of the two results and compare the graph 
with Fig. 27. 

4. A steel ball 10 millimetres in diameter was placed in a 
concave lens of 7 '709 centimetres radius and given a gentle 
push. It was found to make 10 oscillations in 6 '5 seconds. 
What was the periodic time ? 

The periodic time can be calculated from the following 
formula : T= 277^0-4347?. 

Where R=the radius of the concave surface minus the 
radius of the steel ball and all measurements are taken in 

Does the periodic time as calculated agree with the time as 
determined from the experiment ? Carry out similar experi- 

5. A spiral spring had a weight Z placed on the end and was 
given a pull to set it in oscillation. The spring made 48 
oscillations in 10 seconds. Determine the periodic time. 
After completing the experiment it was found that F Ibs. 


X ffx32-2 

extended the spring To feet. If R= _*_ (poundals), deter- 

*4. 12 


Z=51bs., J=0'5 Ibs., and z=0'05. Does the calculated 
periodic time agree with the experimental time ? 

6. A "IT" tube containing water. to a depth of 6 inches 
was slightly displaced from its mean position. The water 
was set in oscillation and made 20 oscillations in 11 seconds. 
What is the periodic time ? Calculate the periodic time from 

the formula T= -r-^/ a where a is taken in feet end repre- 
sents the depth of the water. 

7. Weights of 10, 20, 50 and 100 grammes were suspended 
by equal lengths of cotton (44. cms.). The following table 
gives the time of swing in each case. Determine the time of 
swing and state what conclusions could be drawn if a number 
of other experiments gave similar results. 

Weight in grammes 10 20 50 100 

No. of swings ... 42 42 42 42 
Time to complete swings in 

seconds ... .60 60 60 60 

8. The swings per minute of a pendulum were recorded, 
and are as follows : First minute 44 swings, second minute 
44 swings, third minute 44 swings. It was noted that the 
arc of swing became smaller. Is the time of swing altered 
by the length of arc of swing ? Repeat this experiment. 

9. A number of pendulums of different lengths were selected 
and the following results were obtained : 

Length of pendulum in centimetres : 

20 40 60 80 100 

Period of oscillation in sees. : 

0-85 1-3 1-6 1-8 1-88 

Plot a graph of these results and state what appears to be 
the general effect of shortening the length of the pendulum. 


10. Taking the formula, t = 2n I --- , determine the time 

V y 

of oscillation or period of the pendulums in the preceding 
question. Plot a graph of t and *. 

11. Place some water in a " U " tube and by blowing down 
one limb set the water in oscillation. Determine the periodic 
time of the oscillation. Carry out a number of experiments 
and see whether the following formula gives the same results. 

= -j--\/ a where a is the length in feet of the water column. 

12. Take a straight steel spring or a uniform thin strip of 
wood ; fasten one end in the vice. Set the rod in vibration 
and determine the periodic time. Fasten to the free end of 
the rod a small camel hair brush which is just arranged to 
touch a piece of cardboard. Dip the end of the brush in ink 
and move the cardboard rapidly and uniformly along in the 
direction of the length of the spring. Give a sketch of the 
curve produced. Fill a funnel having a narrow outlet with 
sand and suspend as a pendulum. What figure is traced out 
if a piece of cardboard is placed just below the pendulum ? 
Is the figure of uniform thickness ? 

13. Repeat the above experiment but now move the card- 
board at a uniform rate at right angles to the plane of the 
swing of the pendulum. What diagram is traced out in this 
case ? 

14. How many vibrations per minute would a pendulum 
39" long make at a place where g=32 '2 feet per sec. per sec. ? 

15. Determine the length in inches of a seconds pendulum, 
given the value of g in British units as 32-1740 feet per sec. 
per sec. 

16. A clock loses 5 minutes per day ; the pendulum should 
beat seconds. Determine the alteration in length required 
to make the clock keep correct time. Take g as equal to 
32-174 feet per sec. per sec. 



Propagation of Sound. If one watches- the firing of a 
distant gun, the flash of the explosion is seen and after an 
appreciable interval the sound is heard . Since it is known that 
the flash and the noise occurred together, it follows that light 
travels very much more quickly than sound, and that the 
latter, at any rate, does not travel instantaneously. 

If between the gun and the observer there is an obstacle 
(a building for example) the flash of the gun is not seen, but 
the sound is still heard. From this it follows that light 
cannot ordinarily travel round an obstacle, whereas sound 
does do so. 

Lastly, if the ear is placed to the earth it often happens 
that a distant sound is heard which would have been inaudible 
through the air. It is seen', therefore, that sound can travel 
through solid matter (like the earth) as well as, and perhaps 
better than it travels through the air. 

Experiment has shown that matter of some kind is neces- 
sary for the propagation of sound, for if an electric bell is 
hung in the bell -jar of an air-pump the sound reaches the 
outside very readily so long as the bell -jar contains air, but 
as a vacuum is produced the intensity of the sound rapidly 

Vibration. If we observe very closely a piano wire after 
it has been struck it will be seen that it is " vibrating ; " that 
is, it is moving very rapidly from side to side. The note 
or sound which the wire is producing continues to be audible 
so long as the vibration continues, and it is obvious to the 



most casual observer that the volume of the sound is more 
or less directly proportional to the extent of the movement 
of the wire in its vibration. 

Now the beating of this wire against the air sets up a series 
of " waves " similar in many respects to the wave of bumping 
trucks in the train illustration of the last Chapter. 

In other words longitudinal wave motion is set up in the 
air, ,and it is by this means that the vibration is conveyed to 
our ear and the sensation of sound produced. The wave- 
length is certainly long, often several feet, and the fre- 
quency is very high, usually several hundreds per second. 

The Velocity of Sound. We have already seen that although 
sound travels very rapidly its progress is not instantaneous. 
Before considering the measurement of its velocity it is 
advisable to determine whether all kinds of sound travel 
at the same rate. Does a " high " note, for example, travel 
more or less rapidly than a " low " note ? 

If a band is playing at a distance it is obvious that we hear 
the notes some time after they are actually played. Now 
if different notes had different velocities, notes which were 
played together would not be heard together and the harmony 
would be upset. Such is not the case, and it is therefore safe 
to assume that sound of different frequencies has the same 

To obtain a value for the velocity of sound, in air, it is 
only necessary to record the time occupied by sound to travel 
over a measured distance. Thus if we record with a stop- 
watch the interval between the flash of a gun at a known 
distance and the moment when the noise of the explosion 
becomes audible, we have the necessary data. 

Since the sound is propagated by the actual vibration 
of the air particles it follows that if there is any wind it will 
affect the result. Naturally one chooses for such an experi- 
ment a day when there is as little wind as possible, but even 
so it is impossible to ensure perfectly still air. 

This difficulty is readily overcome by having two guns, 
one at each end of a measured line. The velocity of sound is 


then determined in one direction and immediately afterwards 
in the opposite direction. 

Thus any assistance which was afforded to the sound waves 
by an air movement in one determination would be an equal 
hindrance in the other ; hence the mean of the two results 
gives a reliable value. 

It is found that the velocity of sound in air is affected to 
some extent by the presence of moisture, and also by the 
temperature, but it may be taken to be about 330 metres per 
second or 1,080 feet per second. 

The velocity of sound in water may be determined in a 
similar manner by having two boats at a known distance. A 
bell below the surface of the water is struck by a lever motion 
which causes a flash of light at the same moment. The sound is 
received by observers in the other boat with the aid of a 
submerged listening tube. 

It is found that sound travels .through water with approxi- 
mately four times its velocity in air. 

In both the experiments just described it is assumed that 
no time is occupied by the passage of the light between the 
observers. Although this is not strictly true it can be shown 
that light has the enormous velocity of 186,600 miles per 
second, and hence the time occupied by the light in passing 
over a distance of a mile or two is quite beyond the recording 
limit of a stop watch. 

Vibration of a Stretched String. If a weight of a few 
pounds be suspended by a piece of string about a foot long, 
the string, being in tension, takes up the form of a A 
straight line. If it is plucked slightly the weight im- 
mediately tends to restore the straight line, and in ' 
doing so gives the string a transverse velocity which ' 
carries it beyond the required position, and thus a^' 
periodic motion is established. 

The string, which normally occupies the straight line 
AB in Fig. 33, alternately takes up the position ACE \ 
and ADB. The actual distance is, however, relatively 
much smaller than that shown in the diagram. If 
the weight is increased (other things remaining un- FIG. 33. 


changed) the string will be given a greater velocity, that is, 
its " frequency " will be raised. 

The Monoehord. The monochord, shown in Fig. 34, is a 
suitable instrument for investigating the laws governing the 
vibration of a stretched string or wire. It consists of a strip 
of wood (or a box open at one end) about a yard long and a 
few inches wide. This is fastened to a bench, and a wire or 
string, secured at A, passes over a fixed " bridge " B, then 
over a movable bridge C, and finally over a pulley D to 
carry a scale-pan on which a weight W may be placed. Various 
pieces of string and wire should be tried, but avoid very stiff 
wire which it is difficult to bend. The bridges should be 
made of hard wood, or if soft wood is used a piece of steel 
wire should be fixed along the ridge. 

IB \c 

FIG. 34. 
Experiment 20. 

Fit up a monochord, and keeping the load W constant, 
find the effect of altering the length I between the bridges. 
Next keep I constant and vary the load W. Lastly, keep 
load and length constant, find the effect of substituting a 
thinner or a thicker wire of the same material. 

Musical Note. The ear readily detects the difference 
between a noise and a musical note. Physically the differ- 
ence is that a musical note is propagated by a longitudinal 
vibration of regular frequency and wave-length. A noise is 
also propagated by a longitudinal vibration, but both the 
frequency and the wave-length are irregular. 

A musical note possesses three qualities : (i) pitch (i. e., 
the note is " high " or " low "), (ii) intensity (i,e., the not^e 
is " loud " or " soft "), (iii) timbre or quality. 


The pitch of a note depends upon the frequency of the 
vibration causing it. The more rapid the vibration the 
" higher " the note. The intensity of the note depends 
upon the amplitude of the vibration. The conditions deciding 
timbre will be considered later. 

The Musical Scale. In the upper part of Fig. 35 is shown a 
portion of the keyboard of a piano. This section of notes 
is called an " octave " because it contains eight white notes. 
The letters which are 
used to designate these 
eight notes are shown, 
as is also the notation 
used in music. 

If two notes have the 
same frequency they are 
said to be in " unison " ; 
if they have different 
frequencies the ratio of 
the frequencies is called 
the " interval " between 
the notes. 

At the bottom of Fig. 
35 a row of numbers is; 
given which show the 
relative frequencies of 
the eight notes of the 
octave shown. It will 
be seen that the interval between C and D is 27/24ths or 
9/8ths. Between D and E it is 10/9ths and between E and F 
16/15ths. Following on we have intervals of 9/8ths, 10/9ths, 
9/8ths, and 16/15ths. 

Certain intervals are given names. For example, an 
interval of 9/8ths is called a " major tone," while an interval 
of 10/9ths is called a " minor tone." The interval of 16/15ths 
is called a t: diatonic semitone." 

The numbers given express only relative frequencies, and 
these always hold good. The actual frequency of a note 
varies slightly. The frequencies of all notes are fixed when 

FIG. 35. 



the actual frequency of one is given, and for this purpose the 
C nearest the middle of a piano manual is generally used. 
It is called " Middle C." 

In physics the frequency of middle C is generally taken as 
256 vibrations per second. Modern concert pitch makes it 
276. The French standard pitch has 261, while the Stuttgart 
pitch (adopted by the Society of Arts) makes it 264. " Knel- 
ler Hall," a pitch largely adopted by military bands, gives 
midclle C a frequency of 269 vibrations per second. 

Knowing the pitch of one note and the interval between 

this and another note it is easy to find the pitch of the latter. 


Thus the frequency of G (concert pitch) is x 276 = 414 
vibrations per second. 

The scale just described is called the " Natural Scale." 
A slightly modified scale is now used in music. This is called 
the " Tempered Scale." It makes the interval of a " half 
tone " the same in all parts of the scale. Thus the following 
intervals are equal : C and C$ 9 E and F, B\> and B. The 
following frequencies are useful for comparison. 



E If 












426 -6i 480 






287-4 322-5 




483-3 512 

Experiment 21. 

It can be shown that if a wire is in tension the frequency 
(N) of the note which it emits when it vibrates freely is 
given by : 

N =l /- 

21 \J m 

where 1= length of wire in cms. 

T= tension of wire in dynes. 

m=mass of wire per unit length (i.e. grammes per c.m.). 



Using the monochord the student should test the frequency 
of a few notes on a piano or any other musical instrument 
which is available. 

A few trials may be necessary to get the most suitable 
wire and tension. 

Demonstrate that the " interval " of an octave is 2. In 
other words, the frequency of any note is double that of 
the note one octave below it. 





A A A -A A A .A 
V V V V V V V 

/\. A 

FIG. 36. 

Beats. If any two different notes are sounded together it 
is obvious that the particles of air which convey the sound 
to the ear are performing two different vibrations simul- 
taneously. The actual movement of any particle of air. is 
the resultant of two different movements. 

When two waves of nearly the same wave length are 
combined a phenomenon known as " interference " occurs. 

Consider Fig. 36. Since it is difficult to represent a longi- 
tudinal wave graphically, the diagram shows transverse 
waves only, but the student will readily appreciate that what 
is true of one is true of the other, in this respect. 


The graphs A and B show waves of nearly the same wave 
length, A being slightly the longer. It will be observed that 
at the beginning they are " in phase," but owing to the 
difference of wave length the two waves gradually get out 
of phase until at the point P they are in opposite phase. 
That is to say, a particle conveying such a pair of waves 
would at this point be called upon to move equally in opposite 
directions, the result being no movement at all. At Q the 
waves are back in phase again, the longer having completed 
five waves while the shorter has completed six. 

At C is shown the sum of the two graphs A and B. Fre- 
quent points have been selected and the distance of each 
graph measured from the zero line. Distances above this 
line are taken as positive while those below are negative. 
The algebraic sum provides the data for graph C. 

It will be observed that the amplitude (which is responsible 
for the intensity or loudness of a note) is greatly augmented 
when the original vibrations are in phase, but it sinks almost 
to nothing when the vibrations are in opposite phase. 

The effect of this is to produce an unpleasant beating on 
the ears, the note being alternately very loud and very soft. 
Fig. 36 makes it clear that a " beat " or loud effect occurs 
every time the shorter wave gains one complete vibration. 
Hence if we are producing a note of 256 vibrations a second 
and another note is produced which causes one beat per 
second it follows that the frequency of the latter is either 
255 or 257. 

Experiment 22. 

Fix two monochord wires to produce two notes exactly in 
unison. Then move very slightly one of the bridges so that 
beats are produced. What is the effect of moving the bridge 
slightly farther in the same direction ? 

Beats may be produced very effectively by employing 
two tuning forks of the same pitch and slightly decreasing 
the frequency of one by loading one of its prongs. 

This may be done by attaching a small strip of lead or even 
a bead of wax at the free end of one prong. 


Exercises 6. 

1. Give six sketches showing parts of mechanisms which 
are vibrating in equal intervals of time. 

2. Give the periodic time in each of the above cases. If 
you do not know the exact periodic time give the average. 
Take the periodic time to be the time taken to complete one 
to and fro movement. 

3. The vibration frequency may be defined as the number 
of periods in one second. State the average vibration fre- 
quency in Question 1. 

4. Take a piece of wood of suitable length and of cross - 
section f " by J" ; fasten one end in a vice. Determine the 
period and frequency of the free end. Repeat the experi- 
ment, taking the same length and the cross-section |" by J". 
What can you deduce from this experiment ? 

5. What do you understand by a vibratory motion ? 

6. Describe the way in which sound is propagated through 

7. Distinguish between longitudinal and transverse vibra- 
tions and illustrate your answer by sketches. 

8. Define frequency, amplitude, and wave-length. 

9. The following figures show the velocity of sound in feet 
per second. Express them in metres per second. 

Velocity of sound in air, 1,080 ; in water, 4,900 ; in wet 
sand, 825 ; in granite, 1,664 ; in iron, 17,500 ; in copper, 
10,378 ; in pine wood 11,000. 

10. Boys are placed at two successive telephone poles and 
one of the poles is struck by a blow. Will the sound trans- 
mitted through the wood and wire be heard before the sound 
transmitted through the air ? Use the figures in the preced- 
ing example to give an approximate result. 

11. Explain how you would perform a series of experiments 
on a monochord and show from the results of your experi- 
ments that the following statements are true :: 


(1) The frequency is inversely proportional to the length 
of the string. 

(2) The frequenqy is inversely proportional to the diameter 
of the string. 

(3) The frequency is inversely proportional to the square 
root of the density of the material of which the string is made. 

(4) The frequency is directly proportional to the square root 
of the tension by which the string is stretched. 

12. Assuming that the middle C has a frequency of 256 
vibrations per second, determine the note emitted when a 
card touches a wheel with 64 teeth revolving at the rate of 
8 revolutions per second. 

13. In a mechanism, one of the wheels has 32 teeth and 
is revolving 16 times per second. If the tip of the wheel 
rubs against a piece of material so that a musical note is 
produced, state what is the note ? How many revolutions 
per second must the wheel make to produce the middle B ? 

14. The following table shows the vibrations per second of 
the sequence of the white notes on the piano commencing 
with the middle C : 

Notes C D E F G A B C' 
Vibrations per second 

256 288 320 341 '3 384 426 '6 480 512 

What is the interval between each of the notes as compared 
with C ? Express your results as vulgar fractions expressed 
in their simplest forms. 

15. If the middle C has a frequency of 256 show the posi- 
tion on the treble musical stave of notes having the following 
frequencies : 320, 384, 512. 

16. The humming noise which accompanies the flight of 
certain flies is caused by the beating of their wings. The 
following information is supplied : The wings of a gnat make 
50 per second beats, of a wasp, 110, and of the common house 
fly, 330. From your own experience state what is the 
approximate note in each case and explain how the figures 
supplied bear on your answer. 


17. It has been noted that a series of taps will blend into 
a musical note when their number exceeds 20 per second. 

Using this information state which of the following machines 
will be likely to cause a musical note : 

Ingersoll " Eclipse Rock Drills " 

Size and type . . . B2 C6 E3 FA 

Strokes per minute . . .500 375 350 300 

Petrol engines 

Maximum strokes per minute of the valve tappets 

725 925 625 1050 

18. A wheel with 64 teeth revolves at 4 revolution^ per 
second and a card just touches the wheel. What note is 
produced ? If some of the teeth are now broken off in an 
irregular mariner, state how the sound produced will be 

19. What is the difference between a musical sound and a 
noise ? 

20. A gas engine discharges its exhaust gases at a pressure 
of 25 Ibs. per sq. in. above atmosphere into a long exhaust- 
pipe 65 feet long. Oscillations or waves of pressure are set 
up in the exhaust-pipe. If the engine exhausts 80 times per 
minute, give a sketch of the pressure wave and state what 
will be its frequency. 

21. Discuss the use of the " silencer " on a motor cycle. 
Why is there a difference in the noise produced by the exhaust 
when the " cut-out " is used ? 



Nbdes and Loops. When a length of wire is vibrating 
there are certain points which do not move at all and certain 
others which move a maximum amount. The former are 
called " nodes " and the latter " loops." In Fig. 33, which 
showed the simplest form of vibration, it is seen that we 
have nodes at the points where the wire crosses the bridges 
and a loop at a point mid- way between the bridges. 

Whatever else happens it is obvious that nodes must occur 
at the points where the wire crosses the bridges, but they 
may occur elsewhere also. For example, it is quite possible 
by gently touching a wire midway between the bridges and 
bowing it (with a violin bow) at a point about one quarter 
of its length from one bridge, to induce it to vibrate in a 
manner shown in Fig. 37. 

If such an effect is produced the note emitted is found 
to be an octave above that given by the simple vibration 
(which is called the " fundamental "). In other words, the 
frequency of this note is exactly double that 
of the fundamental. 

It is possible to produce a vibration in 
the wire similar to that shown in Fig. 38, 
in which there are three additional nodes. It 
is found, however, that the conditions for 
this vibration are much more difficult to 
obtain than those in the previous case. It 
is readily seen that other vibrations would 
be possible if the conditions for producing 
them could be obtained. 

Now it has been shown that a stretched 
FIG. 37. wire can vibrate in several ways at once Fiu. 38. 



and in fact always does so. The " note " which is heard is 
usually the fundamental, but more complex vibrations are 
taking place at the same time, although the amplitude of 
these is usually very small indeed compared with the ampli- 
tude of the fundamental. It is for this reason that we do 
not hear as such the notes produced by these small vibra- 
tions, but the sound which they produce goes to make up 
the quality or timbre of the note. 

Harmonics. These small complex vibrations which are 
nearly always present in a note are called " harmonics " or 
" overtones." The vibrations shown in Figs. 37 and 38 
would be called the first and third harmonics (or more cor- 
rectly the " harmonics of the second and fourth orders ") 

If " middle C " be sounded on a piano and the note is 
taken up and produced in exact unison by other instruments, 
such as a violin and a cornet, the note is still middle C, and 
as such its frequency is 256 vibrations per second. The 
three notes, however, are very far from being alike in quality. 

This difference of quality is due to the presence of a different 
set of harmonics in each case. Students who play the violin 
will be familiar with the formation of harmonics. Normally 
a violinist presses a string down with his fingers and the bow 
sets up a vibration in the portion of the string between his 
finger and the violin bridge. This is a fundamental vibra- 
tion. In such cases he obtains a note of higher frequency 
by actually reducing the length of the string which vibrates. 

When, however, the violinist wishes to produce a harmonic, 
he just touches the string at a certain point and bows very 
gently and the whole string is set in vibration : that is, the 
vibration is not confined to the portion of the string between 
his finger and the bridge. 

Resonance. If a very long and very heavy pendulum is 
erected, it requires a considerable effort to set it in motion. 
If, however, it is found either by trial or calculation that the 
period of such a pendulum is, say, 5 seconds, it is possible 
to set the pendulum swinging by applying to the bob a 
comparatively small force at regular intervals of 5 seconds. 


The effect of any one of these impacts is, of course, negligible, 
but owing to the regularity of their application, their cumu- 
lative effect is considerable. If after a while the forces are 
applied at the wrong moment it is possible to counteract 
the effects of those previously applied. 

Experiment 23. 

Erect a heavy pendulum and determine its period in the 
usual way. Having allowed the pendulum to come to rest, 
apply very small taps with one finger, at intervals exactly 
equal to the period. Note the growth of the amplitude of 
the pendulum. Try the effect of a few ill-timed taps. 

Experiment 24. 

Arrange two raonochord wires to give notes exactly in 
unison. If one wire is now plucked, and, after the note 
has sounded for one or two seconds, its vibration is 
stopped, it will be found that the second wire has taken up 
the vibration and is now emitting a note. 
!*r This phenomenon is called " resonance " and its explana- 
tion lies in the principle involved in Experiment 23. In 
this case we have a wire vibrating and sending out in all 
directions waves of sound of a given frequency. These 
waves beat up against the other wire, and although one wave 
is insufficient to make any appreciable difference, their 
cumulative effect is very considerable, as in the case of the 
small forces applied to the heavy pendulum. 

The frequency with which a body freely vibrates is called 
the " natural " frequency, and when impulses reach it having 
a period equal to that of the natural frequency the vibrations 
are said to " synchronise," and when these conditions are 
fulfilled resonance usually takes place. 

Experiment 25. 

Examine the " action " of a piano. It will be seen that 
the striker which sets the wire in vibration when a note on 
the keyboard is depressed withdraws from the wire, leaving 
it free to go on vibrating. As soon as pressure is removed 
from the key, however, a damper is placed upon the wire 
and the vibration immediately ceases. 


If a key is depressed very slowly the striker fails to reach 
the wire and no note is emitted, but so long as the key is 
kept down the damper is raised from the wire. 

Depress middle C in this way, and while it is held down 
strike the C below it, hold it down for a second or two, and 
then release it. Middle C will now be sounding. 

Here we have a case in which the first harmonic of the 
lower C (which is equal in frequency to middle C) has syn- 
chronised with the fundamental of middle C and set the wire 
in vibration. 

Again, having silently depressed middle (7, strike the 
C above it, allow it to sound for a second or two, and release 
it. It will be found that middle C is now emitting a note of 
an octave higher. In this case the vibrations of the funda- 
mental of upper C synchronised with the first harmonic of 
middle C and hence set the wire in vibration of the form 
shown in Fig. 37. 

Note. The student will find a certain amount of mathe- 
matical treatment of harmonics in the authors' Second Course 
in Mathematics for Technical Students, chapter x. Also 
the author's paper on " Vibration of Spars in Aircraft," 
published in Engineering, vol. CIX., page 201, gives some 
practical applications of harmonics. 

Exercises 7. 

1. Show by the aid of sketches what you understand by the 
terms, " node " and " loop." 

2. Show by the aid of a sketch a wire vibrating in such a 
manner as to sound its fundamental note. What difference 
in the sound is caused by the wire vibrating in 2, 3, or 4 
separate portions respectively ? 

3. The same note is sounded on a piano and a violin. 
Explain why it is easy to distinguish the two types of instru- 
ments although the same note has been sounded in each case 

4. What do you understand by the following statement : 
In the case of vibrating strings the frequencies of the sue- 


cessive overtones are as the series of the natural numbers. 
Give drawings to illustrate your answer. 

5. Experiment 21 gives the following formula : 

N= /^ 
21 V ra 

The density of the following materials, in grammes per 
cubic centimetre, is as follows : Aluminium 2 '58, copper 
8 '93, mild steel 7 '68, wrought iron 7 '8, cast steel 7 *9. You are 
given wires made of these materials of the same diameter 
and length and loaded with the same weight. An experiment 
is then performed on the monochord with each of these wires ; 
arrange the metals in sequence, placing the wire which gives 
the highest note at the top. Show that experiment and cal- 
culation give the same result. 

6. A common shop method of distinguishing bars of metal 
is to listen to the note sounded when the bar is struck. If 
equal bars of mild steel, wrought iron and cast steel are pro- 
vided state how you would distinguish them by this method. 

7. In any note sounded by an open organ pipe the harmonics 
may have waves of frequencies of 2n, 3n, 4n t etc., where n 
is the frequency. In a closed pipe only frequencies of 3n } 5n, 
etc., exist. The lack of even harmonics in the latter case 
gives a nasal quality to the tone. Show diagrammatically 
the harmonics in the two types of pipes. 

8. In preparing the foundations of engines it is very 
necessary to arrange that any vibrations arising from want of 
balance should not be transmitted to the rest of the building. 
One method adopted is first to fix concrete foundations for 
the engine, then to place a layer of compressed felt between 
bed plate of the engine and the concrete face. Give reasons 
to show why the felt should help to prevent the transmission 
of vibration. 

9. Show by the aid of sketches how vibration is produced in 
the following cases : (a) by blowing sharply across the end 
of the hole in a key, (6) in an ordinary tin Avhistle. 


10. When two notes are not quite in tune the resulting 
sound is found to alternate between a maximum and a 
minimum of loudness which recurs periodically. Show by the 
aid of a diagram how this is caused. 

11. An aeroplane with two engines is heard some distance 
away and the sound is heard as a succession of beats. It is 
known that the engines are not making exactly the same 
number of revolutions per minute, but the average number 
of revolutions is 2,000 per minute. Give explanations why 
the sound is heard in beats. 



Vibration of Reeds. So far we have only considered the 
vibration of a stretched string (or wire) and it has been 
assumed that the material has been flexible and tension has 
been necessary before vibration became possible. 

By a " reed " is meant a strip of material which possesses 
the property of " stiffness." This enables vibration to take 
place without the material being in tension. 

A reed must always be fixed at one end, and this ensures 
the formation of a node at this point. Whatever form the 
vibration takes there must also be a " loop " at the free end 
of the reed. 


FIG. 40. 

FIG. 41. 

Fig. 39 shows the simplest form of vibration of a reed. 
Figs. 40 and 41 show the forms of the vibrations corresponding 
with the first and second harmonics respectively. 

Relative Frequency of Harmonics. We have already seen 
that the pitch of a note is dependent upon its frequency, 



and tills in turn is inversely proportional to the length of the 
sound-wave. Now, although the wave-length of the sound 
in the air is usually quite different from the length of the 
wave in the vibrating body, the two lengths are directly 

We will now examine Figs. 33 and 37 to 41, and find the 
relative value of the frequency (n) in terms of the length of 
the string or reed (L). 




~ X 

2 L 


Fig. 33^ 



Fig. 37 

1 TT 

1st Harmonic 



2 L 


Fig. 38- 

3rd Harmonic 


2 L 

4 L 






4 L 


Fig. 40 

1st Harmonic 


4 7v 


Fig. 41^ 

2nd Harmonic 



It must be clearly understood that these values of frequency 
are relative and the assumption is made that all conditions 
remain constant. The actual value of the frequency depends 
upon the length, weight, and tension of the string, or the 
length, weight, and stiffness of the reed. 

Suppose the conditions were such that the fundamental 
note in each case was middle C. The first harmonic of the 
stretched string, having a frequency double that of the 



fundamental, would be the C above, usually written 
C'. The third harmonic has a frequency double that 
of C', which takes us up another octave to C". 

Fig. 42 shows the form of vibration of the second 
harmonic of a stretched string, and we see that 

f\ T Q I O 

- and n = ' Hence> if = 24) = 72 ' 

which a reference to Fig. 35 will show is the /* 




note G'. f 


Considering the harmonics of the reed in the same I 


way we may make a table as follows : N, 





Stretched String. 














4 L 

1st Harmonic 






4 L 

2nd Harmonic . 






2 L 


3rd Harmonic 

4 . 






4th Harmonic 






2 L 


Bearing in mind that the " notes " given above refer to 
the harmonics of a stretched string and a reed, both emitting 
middle C as the fundamental, we see that the harmonics 
are quite different, and this accounts for the difference of 
quality or timbre between the notes emitted by these two 

Higher harmonics are of course produced, but sufficient 
are shown in the table to enable the student to extend the 


series if he wishes. The designer of musical instruments 
has to study which harmonics give a pleasant and which an 
unpleasant quality to a note and he then has to foster the 
former and eliminate the latter by suitable design in 
the instrument. 

Experiment 26. 

Obtain a strip of steel (or a steel knitting-needle) and fix 
it in a vice so that the free portion will emit a note when 
made to vibrate. Make a table showing the relation between 
the length of the reed which emits any note and the length 
which emits the octave of that note. (Remember that if 
N is the frequency of any note the octave above has a fre- 
quency of 2 N and the octave below a frequency of \ N.) 

Tuning Forks. The vibration of a tuning fork is only a 
special case of the vibration of a reed. It may be regarded 
as two reeds, attached at one end. Fig. 43 shows - 
the form of vibration. The two prongs alternately 
approach and recede from each other. 

A fork of this nature has two advantages over 
a single reed. Its n&te is sustained : that is, the 
vibration continues much longer than it would in 
a single reed, and unless the vibration is violent 
the note emitted is singularly " pure." In other 
words, harmonics are almost absent. 

Tuning forks of standard pitch may be obtained 
and these form a convenient standard of reference. 
Forks may be set in vibration by gently striking, 
but a better method is to draw a violin bow across FIG. 43. 
one prong. If a fork is placed with its base on a 
piece of wood, or better still a wooden box, the vibrations of 
the fork are conveyed to the wood and the volume of sound 
emitted is greatly increased. 

The Vibration of a Column of Gas. So far we have only 
considered the vibration of solid matter, and its power to 
vibrate has been due to its possessing the property of elas- 
ticity. An elastic body is one which, when deformed by 



an external force, has the power of restoration to its normal 
shape as soon as the force is removed. 

When the properties of gases were being investigated we 
saw that the volume of a given mass of gas varied inversely 
as the pressure. Now, as the gas is compressed by an external 
force and recovers its original volume when that force is 
removed we see that gases are " elastic." That being so, 
they should be capable of vibration like other elastic bodies. 

Experiment 27. 

Arrange eight test-tubes in a stand. On blowing sharply 
across the mouth of any tube a note is emitted. This is due 
to the vibration of the column of air within the tube. If a 
little water is placed in the tube it reduces the length of the 
column of air and the pitch of the note is raised. 

Commencing with the 
^^^ first tube empty place 
water in the remainder to 
such a depth that the musical scale is 
obtained from the eight tubes. Now 
measure the length of each tube above the 
level of the water and plot a graph showing 
the relation between this length and the 
relative frequency of the note emitted. 
(The relative frequencies of an octave are 
given in Fig. 35.) 

Experiment 28. 

Obtain a tall cylinder and nearly fill it 
with water. Support a glass tube, open at 
both ends and about 1 inch in diameter, 
with one end below the surface of the 
water, as shown in Fig. 44. Now, having 
set a tuning fork in vibration, hold it over 
the open end of the tube and raise or lower 
the tube in the water until resonance is 
produced and the tube is emitting a note 
of similar pitch to that of the fork. Measure 
FIG. 44, the length (L) of the vibrating column of gas. 


If forks of other frequencies are available the experiment 
should be repeated, and a graph plotted showing the relation 
between length of column and frequency. 

Now let us consider in detail what is happening in this 
experiment when resonance occurs. Suppose at a particular 
moment the lower prong of the tuning fork is moving down- 
wards. This impact on the adjacent air particles will start 
a compression wave down the tube. This wave will be re- 
flected at the surface of the water and return up the tube. 
For resonance to take place the wave must reach the lower 
prong of the fork as it moves upwards. 

If the frequency of the fork be N vibrations per second 
it will be seen that the interval between the prong moving 

downwards and its commencing to move upwards is -^-^ 

seconds. In this interval the sound wave has moved down 
the tube and up again, a distance of 2 L. 

Now velocity = -1^ 


2 L 

.*. Velocity of sound = 1 

=4 LN. 

This method of determining the velocity of sound is not 
very accurate. A certain amount of disturbance takes 
place at the open end of the tube. A closer approximation 
is given by : 

Velocity=4 N (L+0'4 d). 

where d diameter of tube. 

The student should obtain a value for the velocity of 
sound by this method. It should not be necessary to add 
that all measurements of length should be made in the same 
unit : say the foot or the centimetre. 

The student should note that the column of gas producing 
the sound in the preceding experiment was vibrating with 
a longitudinal vibration, and not transversely as in the case 
of a stretched string or a reed. It is convenient neverthe- 



less to speak of nodes and loops. In the 
case under consideration there was a 
node at the closed end of the tube (i.e. 
at the water surface) and a loop at the 
open end. This is represented in Fig. 45, 
the letters N and L representing node 
and loop respectively. The arrows re- 
present the direction of motion of the 
air particles. 

Organ Pipes. Fig. 46 shows a section 
of a simple organ pipe made of wood. 
FIG. 45. Air passes from the wind chamber 
through the inlet A and passing up the 
channel shown, impinges on the lip B. This causes 
the column of air within the pipe to vibrate, but 
unlike the column of air in the tube in Experiment FlG - 46 - 
28 this pipe is open at both ends and there must 
therefore be a loop at both ends of the pipe. The form of 
vibration for the fundamental note is shown in Fig. 47. 
The forms of vibration for the 1st and 2nd harmonics are 
shown in Figs. 48 and 49 respectively, and it will be noted 
that there is a loop at both ends of the pipe in all cases. 

Closed Pipes. The organ pipe just considered is known as 
an " open pipe." On a modern organ, however, there are 





FIG. 47. 





















FIG. 48. FIG. 49. 


























FIG. 50. FIG. 61. FIG. 52 


always a certain number of pipes exactly similar to Fig. 46, 
but having a tightly-fitting stopper driven into the top. 
These are known as stopped or closed pipes. 

The effect on the nature of the vibration is to make a 
node at the closed end of the pipe. The form of vibration 
for the fundamental is shown in Fig. 50, while Figs. 51 and 
52 show the forms of vibrations of the first and second 

Letting L represent the length of the pipe we may now 
compile a table similar to that referring to the vibration of 
stretched strings and reeds. 




V\ a 4-7 ^ 



lUg. 4H 

TTio- 48 

Open organ H 

1st Harmonic 



JDlg. *O 

Fis- 4<}J 



2 L 

Fiff 50 1 

f Fundamental 


2 L 

Fie 51 

. Closed organ . 

1st Harmonic 



Fie 52- 


w 2nH TTflTmonio 



x g. o^ 



It will be observed that the frequency of a closed pipe is 
exactly half that of an open pipe of the same length, that 
is, the note emitted is an octave lower. On an organ an 
8 ft. " closed " pipe gives a note of the same pitch as a 16 ft. 
" open " pipe. It will be seen from the table, however, 


that the harmonics differ, and hence the note, although of 
the same pitch, is of quite different quality, as every organist 

Exercises 8. 

1. A piece of thin wood was fixed in the vice and set in 
vibration. This experiment was repeated 4 times for the 
following lengths : % 0", I' 6", 1' 0", 6". 


If the frequency =-rr=-(7, 

where =the length, and C equals a constant, determine the 
frequency ratio between the 2'-0" and the other three bars. 

2. Give a sectional sketch of a mouth-organ and mark 
clearly the vibrating tongue. 

3. Examine, if possible, an oboe, a bassoon, and a clarinet. 
Give sectional sketches showing diagrammatically the air 
current and the reed. 

4. If the vibration number of the lowest note on the piano, 
A 4 , is 27, and the speed of sound in air is 1,130 feet per second, 
find the wave-length of the note as it travels through the air. 

5. A thunderclap was heard 4} seconds after the accom- 
panying lightning- flash was seen. How far away did the 
flash occur ? 

6. On a still day the human voice may be heard for 150 
yds. and rifle fire for 5,300 yds. Assuming the usual velocity 
of sound in air state what time sound will take in travelling 
the distances given. 

7. A flat disc has 30 holes, spaced equally, with the centres 
of the holes on the same circle. If the disc is rotated rapidly 
and a jet of air is blown through a small glass tube and made 
to impinge on these holes, state what will happen : 

(a) When the disc runs at a uniform speed. 

(b) A variable speed. 

(c) Double the speed in (a). 


8. In the above question state how many revolutions per 
minute the disc must make to give the following notes of 
standard musical pitch : 

Note C' D' E' C" 

Frequency per second . . 261 293 328 '9 522 

9. Explain why the sirens used in steamers and manu- 
facturing works often give sounds which vary between a 
low note and a shrill shriek. 

10. Explain, by the aid of sketches, why musical sounds 
may be produced on a tin whistle and not when air is blown 
through a parallel piece of tubing of the same diameter and 
length as the whistle. 

11. Describe an experimental method for obtaining the 
number of vibrations per second made by a tuning fork. 

12. Explain why the pitch of the sound rises as water is 
poured into a deep vessel. 

13. On starting an electric motor it will be noted that the 
pitch of the sound gradually rises. Why is this ? 

14. Make a sketch of a bugle. Mark off the length of the 
air column. Explain why this instrument, which has a 
fixed length, can produce several notes. 

15. Make a sectional sketch of the mechanism for the 
formation of sounds in a gramophone. 

16. The velocity of sound in air (V) depends on the 
temperature and may be calculated from the formula : 

= 330 y'l _)_ 0'004. Where V is measured in metres per 
sec., t is the temperature in degrees Centigrade. Calculate 
V for 10, 25 and 30 C. 



Propagation of Light. It is common experience that a 
body exists in the direction in which we " see " it, that is, 
in the direction of the rays of light which pass from the 
body to our eye. From this it follows that light travels in 
straight lines. 

Light travels by means of a wave motion, and in this 
respect is similar to sound, but the wave motion which 
transmits light differs in three important particulars from 
that by which sound is conveyed. 

(1) It is a transverse wave motion, whereas sound waves 
are longitudinal. 

(2) It can traverse a vacuum, which sound waves cannot do. 

(3) The waves are very much shorter, and the motion 
very much quicker than is the case with sound. 

The form of transverse wave motion was shown in Fig. 31. 
The wave length of light depends upon its colour, red light 
having the longest and violet the shortest wave. Yellow 
light, which is of medium wave -length, has a wave which is 
rather less than 0*00006 cms. long. In other words, there 
are about 43,000 complete waves of yellow light in one inch. 
It has already been mentioned that light has a velocity of 
about 186,600 miles per second. 

We are able to see the sun, moon, and stars by means of 
the light which proceeds from them. We know, however, that 
the earth's atmosphere is confined to a comparatively shallow 
layer. Away out in space we have the absence of all forms 
of matter, that is, a vacuum. Yet light passes through it 
quite readily. 

Hence, whatever it is that conveys the transverse wave 




motion of light, it is not matter. Physicists call it " the 

"Shadows. Since light travels in straight lines, it cannot 
pass round a large obstacle as can sound. This accounts 
for the formation of shadows. It is generally found that 
the shorter the wave-length of a vibration the greater the 
difficulty it has in passing an obstacle. 

Consider a source of light which is a point, as $ in Fig. 53 
(an electric arc or a lime-light nearly fulfils this condition). 
Light passes from this point in straight lines, but if an opaque 
sphere, as AB, be placed in the path of the rays, a conical 
space will exist behind the sphere, into which no ray of 
light from S can pass. If a screen is provided, the absence 
of light in this space will be denoted by a shadow CD. 

If the source of light is a 
point, as we have supposed, or 
at any rate is very small com- 
pared with the size of the 
opaque object, the shadow will 
be very well defined, that is, 
the boundary between the illu- 
minated part of the screen and FIG. 53. 
the shadow will form a sharp 

contrast. If, however, the source of light has an appreciable 
size (as is usually the case), the shadow is very badly defined, 
there being a fringe of partial illumination around the shadow 

, Consider the 

case shown in 
Fig. 54. Sis the 
source of light, 
and AB the 
opaque object as 
before. The large 
illuminant may 
be regarded as 
being made up 
of a number of 

FIG. 54. 



points. Let us think of two extreme cases, namely, the 
points G and H. 

Rays of light passing from G will fall on AB in such a 
way as to cast a shadow on the screen at CF, while rays 
from H will fail to reach the portion of the screen DE. 
Since these are the extreme cases for the illuminant, they 
must be the extreme cases for the shadows also. 

Now it is easily seen that the portion of the screen CD 
is in shadow for all cases, the portions CE and DF are in 
shadow for some cases only, while the screen outside EF 
is not in shadow at all. 

The dark portion of the shadow ( CD) is called the umbra, 
while the fringe which diminishes in intensity toward the 
odge is called the penumbra. 

It will be seen that if the source of light is larger than 
the opaque object (as is the case in Fig. 54) the umbra is 
formed by a cone converging to a point at K. If the screen is 
placed beyond this point, there is no umbra shown at all, 
but a nebulous shadow in the form of a ring. This is called 
an " annular " shadow. 

Experiment 29. 

The Photometer. Obtain an opaque rod about an inch 
in diameter, and fix it in a vertical position with a small 
screen of white paper or other suitable material a few inches 
behind it. 

If now a small light (for example, a candle) be placed in 
front of the rod, a 
shadow is formed on 
the screen. If a 
second light (say, a 
small electric bulb) 
be placed in front of 
the rod, but not in 
line with the candle, 
another shadow is 

If the lights are 
placed as A and B FIG. 65. 



in Fig. 55, it is possible to have the two shadows side by side 
and just in contact. It is clear from this diagram that the 
part of the screen represented by DF is illuminated by the 
light A only, while the portion DE receives light only from B. 

Hence if the relative position of the lights is so adjusted 
that the two shadows appear of equal intensity, it is obvious 
that the screen is receiving an equal illumination from both 

We can use this fact to compare the relative intensities 
of the two illuminants. 

First let us consider Fig. 56, in which S is a point of light 
from which rays are passing in all directions. If a square 
screen is placed at A, the amount of light which illuminates 
the surface of the screen is conveyed by the rays passing 
within the pyramid formed by joining the point S to the four 
corners of the screen. 

If a screen were placed at B instead of A, it is easy to 
see that the same amount of light would now fall on a screen 
of much larger dimensions. If the distance SB is double 
that of S A the side of the square B will be double that of 
A, That is, the area of B will be four times as great as the 
area of A, and since the same amount of light falls on both 
screens, it is clear that the illumination of B will be only a 
quarter as bright as that of A. 

Law of Inverse Squares. From the foregoing it is clear 
that, other things being equal, the intensity of illumination 


of a screen varies inversely as the square of the distance from 
the illuminant to the screen. 

Thus in Fig. 55, if P 1 is the power of the light at A and 
P 2 that at B, and the distance from the screen to A and B 
is respectively D l and D 2 , we have : 

Candle-Power. The common unit of intensity of a source 
of light is that given by a candle of a certain type. The 
" standard candle," as it is called, is made of spermaceti 
wax, has a diameter of f inch and burns at the rate of 120 
grains per hour. 

As a matter of fact this standard has now fallen into 
disuse for accurate work owing to its being somewhat un- 
certain, the light given depending to some extent on the 
condition of the atmosphere. 

In modern work a form of burner is used which is fed with 
pentane (a very volatile paraffin). The Harcourt pentane 
burner gives 10 candle-power, and this is now used as the 
international standard. 

Ex. 16. In Fig. 55 let the source of light at A be a standard 
candle, and that at B be of unknown power. The distance 
from A to the screen is 56 cms., and from B to the screen 
85 cms. Find the candle-power of B. 

Candle-power of B 85 2 
1 ~ 56" 

Candle-power of B = I - J 

= (1-517) 2 
= 2-3 

The advantage of this type of photometer is that it is not 
necessary to work in an entirely dark room. The presence 
of another light in the room will not affect the result, pro- 
vided that the additional shadow produced does not overlap 


the two which are being compared. A disadvantage of the 
method lies in the difficulty experienced when the sources of 
light are not approximately points. 

Experiment 30. 

The Grease- Spot Photometer. Make a small screen of 
white blotting-paper and drop a spot of oil on it. Erect 
it in a darkened room, and place a light on one side of it. 

Viewed from the side on which the light is placed the 
oiled spot appears dark on a white ground. This is due to 
the fact that, the spot being translucent, the light passes 
through it, while it is reflected from the rest of the paper. 

Viewed from the other side, however, the spot appears 
bright on a dark ground. Assuming that all the light falling 
on the unoiled screen is reflected, and that that falling on 
the oiled spot is wholly diffused, it is clear that the spot 
should appear as bright in the second case as the remainder 
of the screen did in the first case. 

Now place another light on the side of the screen opposite 
to the first light, adjusting the second light until the grease - 
spot is not distinguishable from the rest of the screen. Since 
the illumination must now be equal on both sides of the 
screen it is possible to compare illuminating powers as before. 

If this method were perfect, the spot, having been made 
indistinguishable from the remainder of the screen when 
viewed from one side, should appear so when viewed from the 
other side. Such, however, is seldom the case. This is 
due to the fact that the light is more scattered by reflection 
from the white paper than it is by diffusion through the 

In practice, therefore, one should aim, not at getting the 
grease spot to disappear, but producing a like contrast 
between spot and screen when viewed from both sides. 

Exercises 9. 

1. Light has a velocity of about 186,600 miles per second. 
What is the velocity in centimetres per second ? 

2. Give simple experiments which prove that light travels 
in straight lines. 


3. The following wave-lengths in " tenth metres " (1O 10 
metres) of characteristic bands and lines in the solar spectrum, 
are given : 4861 '496, hydrogen. 4307 "9, calcium. 5167 '5, 
magnesium. 3820 '56, iron. Express these wave-lengths as 
decimals of a metre. 

4.^ If the wave-length is expressed in " seventh metres " 
(10~ 7 metres), express the following wave lengths in decimals 
of a centimetre : 4 '86, 8, 20, and 25. 

5. Give a list of materials grouped under the following 
headings : opaque, translucent, and transparent. 

6. If it is approximately 93,000,000 miles from the earth to 
the sun, determine how long light will be in travelling from 
the sun to the earth. 

7. A candle is viewed through a pin-hole camera. Explain 
why the image is inverted. 

8. A point source of light is S (Fig. 53), and a cardboard 
disc A B, 2" diameter, is placed with its vertical diameter 6" 
from S with (a) the surface of the disc parallel to the wall 
CD, (b) with the edge of the disc as shown in the figure. 
If the horizontal distance between B and D is 3", determine 
graphically the shape and size of the shadow cast on CD in 
the two cases. 

9. By the aid of sketches show how transverse wave motion 
differs from longitudinal wave motion. 

10. Give some common illustrations to prove that light 
travels faster than sound. 

11. A candle flame J" high is placed 4" from a pin-hole 
in a screen. Find the size of the image produced on screens 
which are placed at distances of 3" and 8" respectively from 
the pin hole. 

12. Draw up a table, giving your impression of the sensitive 
sensation of brightness of the following : A piece of white- 
hot iron, a carbon filament, a metallic filament and a gas- 
filled metallic filament lamp, an acetylene lamp, an oil lamp, 
a frosted incandescent electric lamp. The light giving the 


impression of the highest brilliancy should be placed at the 
top of the table. 

13. A ball 1" in diameter is held 2" from a luminous point 
of light. Give a drawing showing the shape of the shadow 
which is cast on a wall 1J" from a vertical diameter of the 
ball. Will there be a penumbra ? 

14. What do you understand by the " law of inverse 
squares " as applied to photometry ? 

15. The Hefner lamp is used as the German standard of 
candle power and the Hefner standard is equal to 0'9 Inter- 
national Candles. A test was carried out on Osram lamps 
and the following results represent Hefner units of candle 
power : 36-3 29'7, I'O, 22'7, 30'7. What will be the value 
of these results in International Candle Power ? 

16. Experimental observation shows that a white surface 
illuminated by one candle at a distance of one foot appears 
equally bright if illuminated by four candles at a distance of 
2 ft., or nine candles at a distance of 3 ft. State what law 
you would deduce from these observations. 

17. The intensity of illumination produced by the Standard 
Candle at the distance of one foot has been adopted as a 
practical unit of intensity of illumination and is called the 
''foot candle." How many "candle metres" are equal to 
one foot candle ? 

18. Intrinsic brilliancy is a term used to express the degree 
of brightness of a light and is usually given in candle power 
per square inch of the luminous area. Molesworth gives 
the following values : Carbon filament lamp, 375 c.p. per 
sq. in. ; metallic filament lamp (vacuum), 800 to 1000 c.p. 
per sq. in. ; metallic filament lamp, gas filled, 3,500 c.p. 
per sq. in.; acetylene lamp, 100; oil flame, 3 '8; gas flame, 
2 '5. What reasons are there for defining Intrinsic Brilliancy ? 
Contrast these results with your answer to Question 12. 



taws of Reflection. No one can witness a game of billiards 
without coming to the conclusion that in rebounding from the 
raised edge of the table the ball obeys definite laws. These 
laws may be called the laws of reflection, and they apply 
equally well to a beam of light falling on a mirror and a 
smoothly rolling billiard ball. 

Mirrors. For light to be reflected under conditions render- 
ing measurement possible, it is necessary for the reflecting 
surface to be smooth, and free from irregularities. A piece 
of thin plate glass carrying a deposit of silver forms a suitable 
mirror. The surface of the silver next to the glass is the 
brighter, but it has the disadvantage of having the glass in 
front of it. For many purposes this disadvantage is not 
serious, especially if the glass is not very thick. 

A beam of sunlight passing through a chink in a blind 
is readily reflected from a mirror, and the approximate 
direction of the reflected beam is indicated by the position 
of the spot of light which the reflected rays cast on the wall. 

This phenomenon is not readi- 
ly adapted to measurement, 
but it is possible to get there- 
from a general idea of the 
laws involved. 

In Fig. 57 the horizontal 
line represents the reflecting 
surface of a mirror. A beam 
of light travelling in the 
. direction BP meets the mirror 
P at P, and the light is deviated 

FIG. 57. in the direction PC. 



P is called the " point of incidence," and a line BP passing 
along the direction of the beam represents the path of the 
"incident ray," while the line PC represents that of the 
" reflected ray." 

A line drawn from the point of incidence at right angles 
angles to the surface of the mirror (as PA) is called the 
" normal." 

The angle between the incident ray and the normal (the 
angle BP A) is called the "angle of incidence," while the 
angle between the reflected ray and the normal (the angle 
CP A) is called the " angle of reflection." 

We are now in a position to state the laws of reflection. 

The Laws of Reflection. (1) The incident ray, the normal, 
and the reflected ray lie in the same plane. 

(2) The angle of incidence and the angle of reflection are 

The following experiment should make this clear. 

Experiment 31. 

Obtain a strip of mirror about one inch wide and two or 
three inches long, a few pins and a sheet of paper. Fasten 
the paper to a drawing-board and by means of any suitable 
support cause the mirror to rest upon its edge with its re- 
flecting surface at right angles to the surface of the paper. 

Place a pin vertically in the 
drawing-board and close to the 
surface of the mirror (as F in Fig. 
58, in which AB represents the 
mirror). Now place another pin 
about two inches from the mirror, 
and to one side of the first pin (as 
C). If now the eye be placed in a 
position such as D, the reflection of /JT"" 1 
the rays of light passing from the 
pin C will cause an image of the 
pin to be seen (apparently behind !> 
the mirror) in the direction DE. 9^ 


Place a pin near the eye (at D) FIG. 58. 


so that it, the pin at F, and the image appear in a straight 
line. By drawing a pencil-line along the edge of the mirror, 
and joining the points CF and FD, we have a record of the 
path of the incident ray and the reflected ray. 

Draw the normal FG at right angles to AB, and measure 
the angle of incidence (CFG) and the angle of reflection 
(DFG). Repeat the experiment for several positions of C. 

Experiment 32. 

Fix a mirror and the pins F and G (Fig. 58) as in Experi- 
ment 31. Place the eye at D. Now take a long pin and 
place it behind the mirror so that the top of it (which should 
be seen above the mirror) is in line with the pin F and the 
image of C. Move the eye from side to side a few times, 
and if the top of the long pin gets out of alignment when the 
eye is moved, the pin should be moved until no such separa- 
tion occurs. 

When this condition is secured, we know that the long 
pin is not only in the direction of the image of (7, but is actually 
occupying the position of that image, as E in Fig. 58. Join 
EF and CE. 

By measurement demonstrate that the distance from D 
to the image at E is equal to the distance from D to the pin 
C measured along the path which the light has actually 
traversed, namely, CF and FD. Again, show that the 
image E is always as far behind the mirror as the pin C is 
before it, and that the line joining the pin C and its image 
E is perpendicular to the surface of the mirror. 

Curved Mirrors. The two laws of reflection given above 
are applicable to all cases, but so far we have only considered 
the case of reflection from a plane (i.e. flat) mirror. We 
will now consider one or two simple cases of reflection from 
a curved mirror. 

Any curved surface may be regarded as composed of an 
infinite number of very small flat surfaces. Therefore, if 
a ray of light falls on a curved mirror, it behaves as it would 
if it had fallen on a plane mirror which was tangential to 
the curve at the point of incidence. 


Concave and Convex Mirrors. Most of the curved mirrors 
with which the student will deal at this stage of his work 
will be portions of the surfaces of spheres, and may be called 
spherical mirrors. If the reflecting surface is on the 
inside of the sphere, the mirror is said to be concave ; if 
the outside of the sphere is the reflecting surface we have a 
convex mirror. 

The arc of a circle shown in Fig. 59 represents a section of 
the reflecting surface of a concave spherical mirror. O is the 
centre of the sphere. It is clear that if P is the point of 
incidence of a ray of light the behaviour of the ray will be 
such as one would obtain from a plane mirror which was 

FIG. 59. FIG. 60. 

tangential to the sphere at the point P, that is, the radius 
OP will be the normal. 

If AP is the incident ray the angle APO is the angle of 
incidence, and on making the angle OP B equal to this we 
have the angle of reflection and the path of the reflected 
ray PB. 

Fig. 60 represents a convex spherical mirror. is the 
centre of the sphere, and P is the point of incidence of the 
incident ray BP. This time we must produce the radius 
OP to A, to obtain the normal. The angles BPA and APC 
are respectively the angles of incidence and reflection, and 
are of course equal as before. 

Experiments and further particulars referring to spherical 
mirrors will be given in Chapter XII. 



FIG. 61. 

Effect of Rotating a 

Mirror. Consider a mir- 
ror AB in Fig. 61 with 
a ray of light FP falling 
upon it. Since PE is 
the normal the reflected 
, ray will pass along PG. 
Now suppose the 
mirror is turned through 
a small angle into a 
new position CD, the 
incident ray remaining 
stationary. PH will now 
be the normal, and the 
angle of incidence will have been increased by an amount 
equal to the angle through which the mirror has been turned. 
Since the angles of incidence and reflection are equal, the 
latter will be increased by an equal amount also. Thus the 
reflected ray now passes along P J. It is easily seen that the 
angle through which the reflected ray has been turned (viz. 
GPJ) is twice the angle through which the mirror was 
turned (viz. A PC). 

Experiment 33. 

Repeat Experiment 31. Then rotate the mirror through 
about 20 without changing the position of the pins repre- 
senting the incident ray. Record the path of the reflected 
ray as before and measure the angle through which the 
reflected ray has been deviated and compare it with the 
angle of rotation of the mirror. Repeat this experiment, 
using a number of different positions of the mirror. Tabulate 
your results. 

Reflection from a Rough Surface. If we place an electric 
light in front of a mirror the rays of light which, after reflec- 
tion from the mirror, enter the eye actually form an image of 
the light itself. If the mirror is replaced by a piece of paper 
of the same size and shape, no image of the light is produced, 
but the paper is " illuminated." 


Now the rays of light passing from the electric lamp to the 
paper are exactly similar to those which fell on the mirror, 
yet the visual effect is very different. 

Owing to the smoothness of the reflecting surface of the 
mirror the reflected rays occupied a relationship to each other 
similar to that which existed before the mirror was reached, 
and hence they entered the eye under conditions very like 
those which would obtain if they had passed directly from 
the lamp to the eye. 

The paper, on the other hand, has a surface very far removed 
from perfect smoothness, and although each individual ray 
obeys the laws of reflection, the angle of incidence at any 
particular point may be anything, depending upon the 
formation of the paper at that point. Hence the reflected 
rays have no relation to each other in any way resembling 
the initial conditions, and thus no image can be formed. 

Exercises 10. 

1. A ball is travelling in a straight line on a plane surface 
and comes in contact with a plane at right angles to the 
first surface. By the aid of diagrams trace the path of the 
ball under the following circumstances : 

(a) The ball strikes the surface normally and returns 
along the original path. 

(6) The ball strikes the surface at an angle of 30 with the 
normal and returns along a path which makes 30 with the 
normal and 60 with the original path. 

(c) As above, but the surface is struck at an angle of 60 
with the normal and it returns along a path making 120 
with the original direction. 

2. A horizontal ray of light strikes a vertical mirror 
normally. Construct a diagram and state which is the 
Incident Ray, the Reflected Ray, the Normal, the angle of 
incidence and the angle of reflection. 

3. Give a clear account of any experiment you have per- 
formed which proves that the angle of incidence is equal to 
the angle of reflection. 


4. A horizontal ray of light comes from a point on a wall 
and strikes a vertical mirror so that the angle of incidence 
is 30. Show, by means of a scale drawing, where the reflected 
ray will hit the wall if the distance of the wall from the mirror 
is 10 feet. 

5. Taking the data as given in the above question, show 
by means of scale drawings where the light strikes the wall 
for the following cases : 

'(a) Mirror and wall parallel, the light striking the mirror 

(6) The ray of light strikes the mirror normally ; the 
mirror is kept vertical, but swung through 30 in a clockwise 

6. Measure the angle moved by the mirror in the above 
case and state whether the following statement is proved 
in this particular instance : " When the mirror rotates 
through any angle, the reflected beam rotates through twice 
the angle." 

7. A light wooden framework is built up in the form of an 
equilateral triangle ABC. Half-way down the side AB a small 
mirror is fixed at D with its plane parallel to the side AC. 
Pivoted at A is a framework A' C', C' B', resting on A C, 
CB, a mirror being fixed to the point A which moves as 
A' C', C' B' is moved. A horizontal ray of light E A strikes 
the mirror at A and is reflected to the mirror at D. State 
the inclination of the mirror at A to the base BC in order 
that the ray of light may leave parallel to the side BC. 

In connection with this question the student is advised 
to examine if possible the construction of a sextant. 

8. The curvature of a concave mirror is equal to one 
divided by the radius. Express as a decimal of an inch 
the curvature of the following concave mirrors : Radius 
of the mirrors to be 12", 8", 200 millimetres and 80 millimetres, 

9. Make scale drawings of the following concave mirrors 
of radius respectively 4 '5", 7*5", 90 millimetres and 70 


10. In each of the mirrors in Question 9 assume that 
a ray of light proceeds from a point f " above and parallel 
to a radius. Draw in every case the ray and mark the angles 
of incidence and reflection. Measure the angles with a 
protractor and give their approximate values. 

11. State the laws of reflection of light and explain with the 
aid of a diagram the formation of an image in a concave 

12. In a lighting scheme for a drawing office, the following 
surfaces were fixed behind the half-watt lamps used for 
lighting : (a) Brightly polished metal reflectors. (6) Iron 
plates coated with aluminium paint on the underside, (c) 
6 with a final coat of light dull-finished buff -coloured paint. 
In the latter case it was found that the favourable char- 
acteristics were freedom from glare and excellent diffusion. 
Give explanation of this. 

13. Give an explanation as to why a beam of sunlight, 
admitted through a narrow slit to a darkened room, is rendered 
visible. What bearing has the following statement on your 
answer : " In ordinary circumstances there are about 25,000 
dust particles per cubic centimetre of air, rising to about 
250,000 in the neighbourhood of large towns." 

14. A ray of light makes an angle of 10 with the horizontal 
and strikes the mirror of a microscope, which is inclined at 
25 to the horizontal. Will the light be reflected vertically ? 
If not, state the angle of the mirror necessary to accomplish 
this. Illustrate your answer by drawings. 

15. What difference would you expect in the character 
of the light as reflected from (a) a conical white enamelled 
iron shade, 10" lower dia., 1 J hole at the top, and 5" vertical 
depth. (6) As above, but made of opal glass. 

16. Explain why light is reflected differently from plane 
and ground glass. 

17. Why is a room with white walls much lighter than a 
similar room with dark green walls ? 


18. A piece of shafting is firmly fixed at one end and 
supported freely at the other. The free end has an over- 
hanging arm attached to it, just outside the bearing. Near 
each end of the shaft, small mirrors are attached and above 
each mirror a reading microscope with a scale is fixed. When 
the scale is illuminated the image of the scale can be seen by 
the microscope. A cross-hair indicates the position on the 
scale. Explain what will happen when the shaft is slightly 
twisted by means of weights attached to the overhanging 

Show approximately by means of sketches how the twist 
of the shaft is magnified. 

19. Show how you would fix a pair of mirrors parallel to 
each other and one vertically above the other, in order to 
form a simple form of periscope. 

20. Describe experiments by which you would demonstrate 
the two laws which show how a ray of light is reflected from 
a smooth surface. 

21. Rays of light strike a horizontal plane mirror at the 
following angles : 10, 15, 30, 45, 60 and 75. Show 
graphically the angle of deviation. Assume that the angles 
given are angles with the horizontal. 

22. In the above question show how you would arrange a 
second mirror in order that the deviated ray may finally be 
deflected from the second mirror horizontally. Make a 
drawing in each case. 

23. A building is surrounded by high walls. Give sugges- 
tions and sketches showing how you would get as much 
reflected light as possible into the rooms. 

24. A number of rooms receive the whole of their light 
from a rectangular well, open to the air. It is noted that 
the walls of the well are made of white enamelled brick. 
Why is this ? 



CONSIDER Fig. 62, in which AB represents a line of men 
marching with uniform speed. When the line reaches 
CD, it will be parallel to A B, and its direction of motion will 
be unchanged. Suppose, however, that in this position the 
extreme left-hand man is about to step into water about a 
foot deep. This will, of course, impede his progress, and he 
will consequently lag behind his fellows. 

When the line reaches EF, half the men will have entered 
the water, but the man at F will have been in it longest, 
and will have lagged farthest behind. That is, the line will 
have been bent as shown, and although the two portions of 
the line will still be straight, the position representing the 
men in the water will indicate a change of direction in the 
forward motion. 

We have already seen that sound does not travel with the 
same velocity in air and 
in water, and it is found 
that while a number of 
bodies are " transparent," 
that is, are capable of 
transmitting light, they 
do not permit the light 
to travel with the same 
velocity. Thus light tra- 
vels more rapidly in 
vacuo than it does in air, 
and again, its velocity is 
greater in air than in 

Hence, a wave-front of 



light passing obliquely from 
air into a block of glass will 
have its direction of motion 
changed, just as the line of 
men were diverted from their 
original path by their oblique 
entrance into the water. 

Experiment 34. 

Refractio n. Obtain a 
block of plate glass about 
3" X 2" X 1", and place it with 
the large face on a horizon- 
tal sheet of paper. Place 
pins at A and B (Fig. 63), the 
latter being in contact with 
the glass. With the eye in the neighbourhood of K, fix two 
other pins F and K (the former being in contact with the 
glass) in such a way that all four pins appear in a straight line. 
By drawing a pencil line round the block of glass, and joining 
the points ABF and K, we have a representation of a ray 
of light passing obliquely through the glass. 

Consider first the passage of a ray from air to glass. AB 
is the incident ray and B is the point of incidence. Through 
B draw CD perpendicular to the surface of the glass. This, 
is the normal, and the angle ABC is the angle of incidence 
as before. 

On entering the glass, the ray of light is deviated from its 
original path AB, in the direction BF. This phenomenon 
is called " refraction," and the angle between the path of 
the ray within the glass and the normal (viz. FBD) is called 
the angle of refraction. 

When the ray reaches F it passes from glass to air, and it 
should be observed that here the deviation is away from the 
normal. The experiment should be repeated several times, com- 
mencing with a small angle of incidence, and increasing this 
angle so long as the path of the ray can be traced. Tabulate 
the angle of incidence, and the corresponding angles of 



The Laws of Refraction. 
A little thought should make 
it clear that the incident 
normal and refracted rays lie 
in the same plane. 

We will now examine the 
relationship which exists be- 
tween the angles of incidence 
and refraction. Consider any 
one case obtained in Experi- 
ment 34, and draw accurately 
the path of the ray as it 
passes from air to glass. Such 
a case is shown in Fig. 64. 

From the point of inci- 
dence, B, mark off equal 
lengths BF and BH along the 
paths of the incident and 
refracted rays respectively. 
From the points F and H 
drop perpendiculars FG and 
HJ on to the normal. Measure these perpendiculars, and 

obtain the ratio ^r- 

12 J' 

Repeat this construction for every angle of incidence taken 
in the experiment. It will be found that the ratio is con- 

Students with a knowledge of the elements of trigonometry 
will readily see that the above-mentioned ratio is the same 
as the ratio of the sines of the angles of incidence and re- 

The Refractive Index. This ratio is called the refractive 
index of the glass or other medium and is generally denoted 
by the Greek letter jx (pronounced " mu "). We may say, 
therefore, that : 

Refractive index = Sine of angle of incidence. 
Sine of angle of refraction. 

FIG. 64. 



If the experiment had been 
conducted under conditions 
which would have allowed the 
ray of light to pass from a 
vacuum to the glass, the result 
would have been very slightly 
different. The difference, 
however, is so small as to be 
wholly negligible for most 
practical purposes. 

Again, the refractive index 
of a substance depends to 
some extent on the colour of 
the light used, being least for 
red and greatest for violet 
light. A very accurate deter- 
mination on a specimen of 

crown glass gave a value of 1'5 137 for the refractive index 
when red light was used, and a value of 1 '5331 when violet 
light was used. The difference it will be seen, is only about 
1J per cent, of the whole. 

Tables of refractive indices usually give the value of yellow 
light, which is about the mean of the extreme values. 

Fig. 65 shows a method of finding the refractive index 
which may be employed instead of the construction given 
in Fig. 64. The path of the ray is traced in and out of the 
glass block. The line representing the ray at exit is then 
produced backwards until it cuts the normal as shown in 

the figure. The ratio 7T ^ is the refractive index. Students 


with a little knowledge of geometry will find it an interesting 
problem to prove that the constructions shown in Figs. 64 
and 65 give the same result. 

Referring again to Fig. 63, the student should observe 
that if the block of glass is rectangular, or if the opposite 
faces are parallel, the ray of light F K is parallel to AB. 
This should be verified by an examination of the various 
figures obtained in Experiment 34. 



Passage of a Ray of Light through a Prism. In Fig. 66, 
ABC indicates a prism of glass whose section is an equi- 
lateral triangle. If a ray of light fall on one face of this 
prism in a direction DE, it will, on entering the glass, be 
refracted towards the normal and follow a path EG. 

At G it passes from glass to air, and it will consequently 
be refracted away from the normal, as C J. 

Now it has already been mentioned that the refractive 
index of a medium depends to some extent on the colour 
of the light used. Hence, if DECJ represent the path of 
a ray of red light, it is clear that if violet light were employed 
the angle through which it would be refracted would in 
both cases be greater and its path would have been similar to 

FIG. 66. 

Now if a beam of white light is passed through a prism in 
the direction of DE in Fig. 66, a band of colour is produced, 
commencing with red at J and passing through the other 
"colours of the rainbow "orange, yellow, green, blue, 
indigo and apparently ending with violet at L. 

This indicates that " white " light (for example, sun- 
light) is composed of many colours, and if light of these 
colours be suitably merged together it is possible to produce 
white light. This band of colour is called the continuous 

Experiment 35. 

Trace a ray of light through a prism of glass by means 
of pins, placing pins at DEG and J, so as to appear in a 



straight line when the eye is placed beyond J. The student 
will note that the pins at D and E appear fringed with colour. 
This is due to the fact that the pins are reflecting more or 
less white light. 

By producing the line DE and producing JG to meet 
it at N, we are able to measure the angle MNJ, which is 
the angle through which the ray has been deviated by its 
passage through the prism. 

Repeat the experiment several times, using different angles 
of incidence. Make a table showing the relation between the 
angle of incidence and the angle of deviation. What are 
the conditions which give minimum deviation ? 

Total Internal Reflection. When light passes from glass 
or water or other similar media into air, we have seen 
that it is deviated away from the normal. That is, the 
angle between the normal and the path of the ray is greater 
in air than it is in water. 

Consider Fig. 67. Let light pass through a point P in a 
block of glass. If it meet the surface of the glass normally, 
as PA, it passes on without deviation, as AE. If it take 
the path PB, it is refracted to BF. (This is a repetition 
of the case of the ray BF K in Fig. 63.) 

Now it is obvious that since the angle FBK is greater 
than the angle PB J there will come a time, as the ray within 

the glass becomes more 
oblique, when the angle 
between the external ray 
and the normal is a right 

This condition is reach- 
ed by the ray PC, which 
is refracted so that it 
G passes along the surface 
of the glass in the direc- 
tion CO. The angle PCL 
is| called the " critical 

If this angle is ex- 



ceeded, as in the case of PD t the 
ray fails to pass into the air and 
" total internal reflection " occurs, 
as DH. 

This mode of reflection of light is 
the most perfect known and is 
utilised in the reflecting prism which 
is shown in Fig. 68. It consists of a 
glass prism whose section is a right- 
angled triangle. 

If light strikes the side of the 
prism, represented by A B, normally, 
it passes into the glass without deviation, 
the ray DE. This light has an angle of 

FIG. 68. 

as shown by 
incidence on 

the hypotenuse EG which is above the critical angle. Hence, 
the whole of the light is reflected in the direction EF, and 
since it strikes the side C A of the prism normally no deviation 
is produced by the exit of the light from the glass. 

The following table showing the indices of refraction of a 
few common media will be useful for reference : 


Refractive Index. 


Boro-silicate crown 



Soft crown ..... 


Hard crown .... 


Barium crown . . - 


Light flint .... 


Dense flint ..... 


Quartz ...... 


Carbon bisulphide .... 




Ice ....... 



Experiment 36. 

Examine a reflecting prism. Trace the path of a ray of 
light through it by means of pins. What happens if the 
incident ray does not fall on the first face of the prism nor- 
mally ? Is it possible to select an angle of incidence so 
that total internal reflection does not occur ? 

Exercises 11. 

1. Give an account of any experiment which you have 
carried out which proves the following general law for refrac- 
tion : When light travels obliquely from one medium into 
another in which the speed is less, it is bent towards the 
perpendicular, and when it passes from one medium to 
another in which the speed is greater, it is bent away from 
the perpendicular produced into the second medium. 

2. A penny was placed at the bottom of a bowl in such a 
position that the edge just hid the coin from the observer. 
Explain why it becomes visible when water is poured into 
the bowl. 

3. Why does a stick held in water appear bent to the 
observer ? 

4. A ring is at the bottom of a pond one foot deep and 
lies two feet from the observer. Give a diagram to scale, 
showing its apparent position to the observer. 

5. A boy holds a penholder at the back of a piece of glass 
and looks through the glass obliquely. Give a diagram to 
show what he sees. 

6. A diver is being lowered into the sea and the life-line 
is held vertically above him. Give a drawing to show how 
the line looks to the diver. 

7. In a house there are two windows, one being made from 
glass having a plane surface and the other being made from 
cheap glass, having a surface which is not quite plane, but 
has small depressions in it. Explain why a person in the 
street looks distorted through one window and not through 
the other. 


8. Explain why the reflections of objects seen in cheap 
looking-glass give distorted images. 

9. The index of refraction may be defined as the ratio of 
the speed of light in air to its speed in any other medium. 
The following refractive indices are given: Water 1'33, 
alcohol 1*36, benzene 1'50, and flint glass 1*67. Calculate 
the speed of light in these media. 

10. Rays of light proceed from air into water, making 
angles of 10, advancing by 10 to 40. Give drawings and 
mark clearly the normal angle of incidence and angle of 
refraction in each case. Take the index of refraction of water 
as 1-33. 

11. Draw any horizontal line XY and on this line con- 
struct a circle of 3" radius. Through the centre R draw a 
vertical diameter A A' t and in the top left-hand quadrant 
draw radii RO, RB, EG, making angles of 10, 20 and 
30 with the normal A A'. Drop perpendiculars from 0, B y 
and C on to A A' . Make three triangles in the bottom 
right-hand quadrant in which the perpendiculars dropped 
from 0', B' ', and C' are equal to the corresponding per- 
pendiculars divided by 1*5. 

If the refractive index for benzene is 1-5, which angles 
represent the angles of incidence and refraction in this con- 
struction ? 

12. Opaque objects, such as metals and alloys, when 
examined by the metallurgical microscope must be examined 
by reflected light. Vertical or oblique illumination may be 
used. To produce vertical illumination the following methods 
have been tried : 

(a) A small annular silver mirror forming an angle of 45 
with the axis of the microscope. 

(6) A semicircular mirror mounted as above and partially 
covering the objective. 

(c) A very small central mirror mounted as in (a). 

d) A totally reflecting right angled prism, covering half 
the aperture of the objective. 

(e) A plain glass disc, 


Assuming a highly polished specimen and a horizontal 
beam of light, show diagrammatically in each case how the 
light is reflected to the eye. The beam of light is admitted 
to the tube of the microscope through a hole in its side, is 
reflected downwards by the reflector, through the objective, 
and thence is reflected back to the eye by the object. 

13. Air changes in density as the pressure and temperature 
vary. When high atmospheric altitudes are reached the 
index of refraction of the air becomes less, owing to the 
diminution in density. Show by diagrams the real and 
apparent positions of a star, (a) at its zenith, (6) just below 
the horizon. 

14. Explain, by means of a diagram, the term " critical 

15. A coin lies at the bottom of a pond which is 2 feet deep. 
Explain by means of a diagram why it appears at a less 
depth to an observer in a boat. Calculate its apparent 
depth from the formula given : 

Real depth of the water. 
Index of refraction =-r 

Apparent depth of the water. 

16. If a body be viewed through a plate of transparent 
material, it appears to be nearer to the observer than it 
really is. Taking the following formula : 

Real thickness of medium 
Index of retraction mi^T" 7~n 

Apparent thickness of the medium 

calculate the apparent depth or thickness of the following 
media, assume that the real depth in each case is 6" : Indices 
of refraction, diamond 2*44, flint glass 1'58, crown glass 
1*5, carbon bisulphide 1'68, water 1*33. 

17. Taking other media, give a simple illustration to 
show that the folio whig statement is likely to be true : " Rays 
of starlight passing through the atmosphere are refracted or 
curved downwards. The effect is to make objects appear 
higher than they really are." 


18. Give a description of the simplest form of heliograph 
with which you are familiar. 

19. During an ordnance survey it was necessary to pick 
out a distant station. A cone of burnished tin was hoisted 
on a pole. Explain how this would help. 

20. A beam of light is directed through rectangular 
vessels with glass sides containing the following liquids : 
water, alcohol, and cedar wood oil. Show diagrammatically 
the path of the beam given the following refractive indices : 
water 1'333, alcohol 1'36, and cedar- wood oil, the same as 



The Formation of Images. Everyone is familiar with the 
formation of an " image " on a screen, for we see it in the 
projection of an optical lantern and its modern development 
the cinematograph. A magnified " image " of an object 
is also seen when one applies the eye to a telescope or micro- 
scope. We will now consider the formation of images, 
first by the reflection, and second by the refraction of the 
rays of light which emanate from the object. 

The Focal Length of a Spherical Mirror. Consider a 
concave spherical mirror indicated by the arc A BC in Fig. 69. 

FIG. 69. 

O is the centre of the sphere. The line passing through 
O and the middle (B) of the mirror is called the axis of the 

If an incident ray DE is parallel to the axis it will be 
reflected along a path EF such that the angles of incidence 
and reflection are equal. EO is, of course, the normal. 




Provided that the arc ABC is only a very small portion 
of the whole sphere (in other words, if the curvature of the 
mirror is not very great) it is found that all incident rays 
which are parallel to the axis are reflected along lines which 
pass through the common point F. This point is called the 
focus, and the distance from the mirror to the focus (namely, 
BF) is called the " focal length " of the mirror. 

In Fig. 70 ABC represents a convex spherical mirror, 
BO is the axis and the centre of the sphere. The incident 
'ray DE is reflected along EJ (OE produced being the 

FIG. 70. 

Now it is obvious that, since these rays are divergent, 
they will never meet in the direction in which they are 
travelling. But if the lines are produced backwards, they 
meet at F. So F is the focus of this mirror and BF is its 
focal length. 

It is necessary to emphasize that this convergence of 
parallel rays to a point only occurs when the mirror is a 
very small portion of a sphere. If this condition is fulfilled, 
the focal length is found to be half the radius of curvature. 
Hence in Figs. 69 and 70, BF is half the length of BO. 

Real Images. In Fig. 71 a concave mirror is shown having 
its focus at F, and the centre of the sphere at O. A bright 
object is placed in front of the mirror, the object being 
indicated in the diagram by the arrow AB. 


We may assume that rays of light are being sent out from 
this object in all directions. Some of these rays will fall 
on the mirror. Consider the ray AG. This, being parallel 
to the axis of the mirror, will be reflected in a direction CD, 
passing through the focus F. Another ray passing from A 
will strike the mirror at the middle point E, and will be 
reflected along EG (the angle of incidence AEO being equal 
to the angle of reflection GEO). These lines intersect at 
H, a^id if 'other rays from A were drawn, careful construc- 
tion would show that they are all reflected along lines 
passing through H. 

H, we may therefore assume, is the position of the arrow- 
head in the image. The ray BE, since it strikes the mirror 

FIG. 71. 

normally, will be reflected back along its own course. The 
position of the tail of the arrow is therefore obviously at J. 

Experiment 37. 

A 4- volt electric bulb forms a very convenient " object " 
for experimental work in optics. Failing this a light may 
be enclosed in a box and the rays allowed to 'escape only 
through a hole about half an inch square, the hole being 
covered with wire gauze. 

Various forms of mirror- holders are purchasable, but it 
should not take long for an ingenious student to see the 
possibilities of a cork and a piece of plastic wax or clay. 
A suitable screen consists of a piece of white card about 
3 inches square. It may be mounted in a split cork. 

Now place the " object " on a table in a darkened room 
and place a concave mirror about a foot in front of it. Move 
the screen to and fro until the " image " is seen. If an 



electric bulb is being used the image should be the filament 
of the lamp. By a careful adjustment of the screen the 
image may be " focussed," that is, it may be made to appear 
quite sharp and well-defined. 

It is obvious that the screen must not obstruct the rays 
of light passing from the object to the mirror, hence it is 
necessary slightly to twist the mirror so that its axis passes 
between the object and the screen. 

Measure the distance from object to mirror, and from 
mirror to image. Also measure the length of the object 
and image. Repeat the experiment several times, placing 
the object at a different distance from the mirror in each 
case. Tabulate the measurements mentioned above for 
future reference. 

The Spherometer. This instrument is employed for the 
determination of the radius of curvature of spherical sur- 
faces. It consists of a very small steel tripod whose feet 
form an equilateral triangle. A fourth foot which forms 
the base of a micrometer screw passes through the middle 
of this triangle. 

The instrument is first placed on a sheet of plate glass, 
and the micrometer adjusted so that the central (movable) 
foot just touches the glass. It is then 
transferred to the spherical surface 
and the micrometer again adjusted for 
contact. The difference in the micro- 
meter readings gives the difference 
between the level of the tripod feet 
and that of the micrometer foot, due 
to the curvature of the surface. 

In Fig. 72, let ADE represent the 
curved surface. Let B represent the 
point of contact of one of the tripod 
feet, and let D be the point of contact of the micrometer foot. 

GD= difference of micrometer readings. Denote this by h. 

C B= distance from a tripod foot to the micrometer foot, 
which may be measured on the spherometer. Denote 
this by d. 


0jB=radius of the sphere. Denote this by R. Now'OD 
is another radius, and hence OC = (R h). 

In the right -angled triangle OCB we have : 

i.e.R 2 =R 2 2Rh+h 2 +d* 
Hence 2Rh =h 2 +d 2 
h 2 +d 2 

and R = 


It will be recalled that we have confined our consideration 
of spherical mirrors to those having very little curvature. 
Hence in applying a spherometer to them, h will be a very 
small quantity, and in most cases the term h 2 may be omitted 
without introducing any appreciable error. The formula 

in that case becomes : R = ^r 


Experiment 38. 

Measure the radius of curvature of the mirror used in 
Experiment 37. Use the formula containing the term h 2 , 
and then find the percentage error introduced by omitting 
this term. 

Returning to the table of results obtained in Experiment 
37, let the distance from the object to the mirror be called 
u, and the distance from the mirror to the image be called v. 
Then denoting the focal length of the mirror by / it can be 

, shown that : -- = 1 

/ u v 

The focal length of the mirror should be calculated for 
each case, and the mean of these results obtained. We have 
already seen that the focal length of a spherical mirror is 
half its radius of curvature, so that the focal length obtained 
by the optical method may be compared with that given by 
the spherometer measurement. 

When an image can be thrown on a screen, it is said to be 
a "real" image. In Fig. 71, the image may be described 
as : " real, diminished and inverted." 


Virtual Images. We will now consider the behaviour of a 
convex mirror. This is shown in Fig. 73. Again AB repre- 
sents the object, and F the focus. The ray A C being parallel 
to the axis is reflected in the direction CD and the ray AE 
passes along EG (the angle AEB being equal to the angle 

Now these two reflected rays are divergent, and will 
therefore meet only if produced backwards. The intersection 
takes place at H, which determines the position of the head 
of the image. 

Since this image is behind the mirror, it is obviously 
impossible to throw it on to a screen, and hence it cannot 
be called " real " in the sense that the last image was. 

This is a case of a " virtual " image. If the eye is placed 
in front of the mirror, the virtual image is seen, and in the 
case illustrated in Fig. 73 the image may be described as 
" virtual, diminished and erect." 

Lenses. A lens is a piece of transparent material (usually 
glass) having a curved surface (usually spherical). Fig. 74 
shows the form in section of a number of common lenses. 
In A both surfaces are convex, and such a lens is called a 
" bi-convex lens." In B both are concave, and hence it is 
called a " bi-concave lens." In C and D one surface is 
flat (or plane) and the other spherical. C is a " plano- 
convex lens," and Da" plano-concave lens." In E and 
F both surfaces are spherical, but one is concave and the 
other convex, 



In our consideration of lenses we shall assume that in all 
cases the curvature is very small, and the lens itself very 

Of course there are lenses which do not fulfil these con- 
ditions, but their behaviour is somewhat complex, and the 
study of such lenses is best delayed until the student has 
had experience with those of the simpler type. 

f\ U 

1 1 

z r 


FIG. 74. 

Lenses are conveniently divided into two classes: (1) 
Those which cause a parallel beam of light to converge are 
called convergent lenses. These are thicker at the axis 
than they are at the edge of the lens. A, C, and E, in Fig. 
74, are convergent lenses. Those which are thickest at the 
edge, such as B, D, and F, in Fig. 74, are called divergent 
lenses, because they cause a parallel beam of light to diverge. 

The Focal Length of a Lens. In Fig. 75 A B represents a 
bi-convex lens, which is, of course, convergent. A line drawn 
through the centres of the spheres of which the curved sur- 
faces of the lens are portions is called the axis of the lens. 
CD and EG represent rays of light whose paths are parallel 
to the axis of the lens. 

FIG. 76. 



When these rays pass into the lens they are refracted in a 
manner similar to that in which a ray is refracted by a prism. 
It is found that all such rays pass through a point F on the 
axis, called the focus of the lens. The distance from the 
lens to the focus is called the focal length of the lens. 

In Fig. 76, AB represents a divergent lens. In this case 
the rays CD and EG, which are parallel to the axis, are re- 
fracted by the lens so that their paths diverge as shown. 
All such rays, however, on being produced backwards pass 
through the common point F, which is the focus of this lens, 
and again the distance from the lens to the focus is called the 
focal length. 

Strictly speaking, part of the deviation of a ray of light 
passing through a lens takes place at the front surface and 
part at the back surface of the lens. This is not shown in the 

Images Formed by Lenses. In Fig. 77 let AB represent a 
convergent lens of which the focus lies at F. Let CD be an 
object. Consider 
a ray passing 
from C parallel 
to the axis. This 
will be refracted 
through the 
focus. Another 
ray from C is 
shown passing 
through the mid- 
dle of the lens. 
If, as we have assumed, the lens is thin, and not deeply 
curved, this ray suffers practically no deviation, but passes 
on in a straight line as shown. 

These two rays meet at E, and this is the position of the 
head of the image. 

Experiment 39. 

Mount a convergent lens, a screen and an " object " in 
a darkened room as in Experiment 37. The screen and 


object must, of course, be on opposite sides of the lens. 
Again measure the distance u (from object to lens) and v 
(from lens to image) for a number of cases. If / is the focal 
length of the lens calculate its value in each case and obtain 

the mean. The formula : r = 1 

/ u v 

is still true as in the case of the concave spherical mirror. 

Experiment 40. 

Having obtained the focal length of a convergent lens in 
Experiment 39, take the same lens into the sunlight (which 
may be regarded as parallel rays of light). These rays 
converge to a point, which appears as a bright spot of light. 
This is, of course, the focus of the lens, and the focal length 
may, therefore, be measured directly. 

Since the lens concentrates the sun's heat rays as well as 
the light rays, the spot of light is likely to be hot. The 
student should have no difficulty in testing whether this 
is so. 

Experiment 41. 

Using the same lens, hold it near the wall of a room opposite 
the window. With a little adjustment an image of the 
window is obtained on the wall. This is very similar to the 
darkened room experiment except that the distance u is 

very great, and consequently is . very small indeed and 
may be neglected without serious error. 

Hence the formula becomes : *- = - - and / = v. 

f v 

The distance from the lens to the wall is, therefore, the 
focal length of the lens. 

The student is now in a position to verify the fact that the 
ratio of the sizes of object and image is the same as the ratio 
of their respective distances from the lens or mirror. In othei 

words the " magnification " of the image is equal to . 



FIG. 78. 

If this fraction is less 
than unity the image is 
diminished. A magni- 
fied image will be indi- 
cated by this ratio being 
greater than unity. 

A divergent lens is 
indicated by A B in Fig. 
78, the focus of which 
is shown at F. CD is 
an object as before. The ray from C which is parallel to 
the axis is refracted in the direction EG (note that GE 
produced passes through F). The ray from C which passes 
through the middle of the lens proceeds in the direction of 
H without deviation. 

Intersection takes place at */, and J K is therefore the 
image. Note that it is virtual (and therefore cannot be 
thrown on a screen, but if the eye be placed between G and 
H it can be seen), diminished and erect. 

The Magnifying Glass. Referring back to Fig. 77 it will 
be seen that the distance from the object to the lens was 
greater than the focal length of the lens. We will now find 
the effect of making this distance less than the focal length. 

Fig. 79 shows the lens AB with its focus at F. CD is 
the object. Using the same construction we find that the 




FIG. 79. 


rays which before converged to form a " real " image are 
now divergent and the virtual image GH is produced. Note 
that this image is magnified and erect. 

If the eye be placed between F and E this magnified 
virtual image is seen, and this is the explanation of the 
magnifying glass or simple microscope. 

Exercises 12. 

1. Draw a concave mirror of 4" radius, and with a 2" arc. 
Mark on the drawing (a), (6), and (c) as defined. 

(a) Centre of curvature is the centre of the sphere of which 
the mirror forms part. 

Bisect the arc of the mirror and join to the centre. The 
point where the radius cuts the arc or middle point of the 
mirror is called the pole (6). 

(c) The principal focus is midway between the centre of 
curvature and the pole. The focal length is the distance from 
the pole to the principal focus, and if the focal length of the 
mirror is called / and the radius of curvature r, insert measure- 
ments in your drawing and show that r=2f. 

2. Repeat the above example for concave mirrors of 3", 
3J", and 5" radius with 2" arcs. 

3. Take concave mirrors of 2 '75", 3 -25" and 3 '375" radius 
and in each case with arcs 2 '5" long show where rays 
parallel to the principal axis intersect the principal axis 
after reflection. 

4. A luminous point O is situated on the principal axis of 
the mirrors in the preceding question and lies 1" behind the 
centre. A ray of light proceeding from this point strikes 
the mirror 1" above the principal axis. Show graphically 
where the reflected ray intersects the principal axis. 

5. The following information is supplied in a catalogue 
with regard to a series of concave mirrors : 

Diameter in millimetres . 50 63 75 100 150 
Focal length in millimetres 75 75 80 100 180 

Determine graphically in each case the radius of curvature. 


6. Similar information to that of the preceding example is 
supplied in respect of convex mirrors : 

Diameter in millimetres . . 50 63 75 100 
Focal length in millimetres . .75 75 80 100 

Find the radius of curvature in each case. 

7. Measurements were taken of a number of spherical 
watch-glasses and the following readings were noted : 

Spherometer readings Chordal diameter 

in mm. in cm. 

Case 1 . . 4-26 6'4 


Case 2 . . 6-88 5 

Case 3 . . 1'49 10 


Case 4 . . 1'99 11 


Taking the distance G B as 2 '25 cms. calculate the inside and 
outside radius of the sphere which forms the watch-glass. 
Make a scale drawing in each case. 

8. In case 1, 3 and 4 of the above question, the inside of the 
watch-glass was blackened. The outside curve was then used 
as a convex mirror. Show graphically how a ray of light 
parallel to the principal axis is reflected in each case. 

9. Make sketches of any spherometer you have used and 
name the different parts. 

10. The following results were obtained with a spherometer : 
Distance from tripod foot to micrometer foot, 3*1 cms. 
Difference of micrometer readings for a concave mirror, 
0*15 cms. ; difference of micrometer readings for a concave 
mirror, 0'127 cms. Calculate the radius of curvature for 
each mirror. 

11. Make sketches of reflectors used for bicycle, motor car 
and railway lamps and electric light. Give sectional views 
wherever possible and state what are the approximate shapes. 


12. The inside of each of the watch-glasses in Question 7 
is blackened and they are then used as convex mirrors. 
Show how a ray of light 1 cm. from the principal axis is 
reflected in each case. Neglect the slight difference of 
curvature between the two surfaces. 

13. A lens which changes a parallel beam of light into a 
convergent beam is caUed a " converging or convex lens." 
These lenses are always thickest in the centre. Draw and 
name three types. 

14. A lens which changes a parallel beam of light into a 
divergent beam is called a diverging or concave lens. These 
lenses are always thinnest in the centre. Draw and name 
three types. 

15. Define the " focus " of a lens. 

16. An experiment was performed to find the focal length 
of a convex lens. The lens was fixed and a pin was placed 
at some distance from it Another pin was placed on the 
other side of the lens until the image and the pin-head -ap- 
peared together. The following results were obtained : 

u, distance from object 49 '2 cms. v, distance from image 

22-5 cms. 
35-6 25 cms. 

Determine the focal length of the lens. 

17. This experiment was then performed with the object 
nearer to the convex lens than the principal focus. Give 
drawings, showing the image of the pin in this case. Is the 
image real or virtual ? 

18. An experiment was performed to determine the focal 
length of a convex lens. The lens was placed in front of 
the screen and a strongly illuminated pin was placed at the 
other side of the lens The lens was moved until a sharp 
image was obtained on the screen. The following results 
were obtained : 

u y 42 cms., u, 61*5 cms. v, 67 cms., v, 43*3 cms. 

Determine the focal length of this lens. 


19. Describe, with the aid of sketches, why the image of 
a pin as seen in a plane glass mirror differs from that seen on 
the ground-glass screen of a camera. 

20. Why is a magnifying glass sometimes called a " burn- 
ing glass " ? Show how you would find the principal focus 
of a Ions by means of the sun. 

21. An arrow 1J" long is 2" from the pinhole in a pinhole 
camera. If the arrow is vertical, show by means of a diagram 
the image formed on a screen 3" away from the pinhole. 




Heat and Cold. From our early days we are accustomed 
to associate certain sensations with heat and cold. Such 
sensations are, however, not very trustworthy, for in certain 
circumstances two men may not be in agreement about the 
heat (or coldness) of a body. 

Experiment 42. 

Take three basins, and into No. 1 put hot water. In 
No. 2 put some ice-cold water and fill No. 3 with lukewarm 

Now place the left hand in the hot water in No. 1, and 
the right hand in the cold water in No. 2, and leave them 
there for one or two minutes. 

Then put both hands into the tepid water in No. 3. 
This water will feel cold to the left hand and warm to the 
right hand. 

This experiment shows how unreliable is the sense of 
touch as an indication of heat or cold. 

Because the left hand was hotter than the warm water, 
heat flowed from the hand to the water and the hand felt 
the loss of heat as a sensation of coldness. The right hand 
being colder than the water, heat flowed into it from the 
water, and this acquisition of heat gave the sensation of 

Temperature. The degree (or intensity) of heat (which is 
quite distinct from the amount of heat) is commonly denoted 
by the term " temperature." 




Experiment 43. 

Obtain a cylindrical rod of metal with trued ends, also 
a piece of sheet metal cut to the form shown hi Fig. 80, the 
dimension of which are such that the rod will just pass into 
the gap in the sheet metal and the end of the rod will just 
enter the circular hole. 

Now hold the rod in the flame of a Bunsen burner for a 
minute or two. It will be found that it will neither pass 
within the gap nor enter the hole while it is hotter than the 


FIG. 80. 

sheet metal, but will do so when it has cooled to approxim- 
ately the same temperature again. 

The two foregoing experiments establish two facts : 
(1) When two bodies, of different temperatures, are in con- 
tact, heat flows from that at the higher temperature to that 
at the lower temperature, with a tendency to establish 

(2) The dimensions of a body depend upon the tempera- 
ture. In most cases the dimensions increase as the tem- 
perature rises. 

It is obviously quite possible (though perhaps not very 
practicable) to use a rod of iron as a thermometer. Since 


the length of the rod is a function of its temperature, all 
that is necessary is to measure the length of the rod and 
determine the temperature corresponding to this length. 

This method has two very grave disadvantages : (i) 
the iron rod in absorbing heat necessarily reduces the tem- 
perature of the body whose temperature it is required to 
determine ; (ii) the increase in length for any reasonable 
rise of temperature is so small that accurate measurement is 
extremely difficult. Moreover, the measure will be affected 
by heat. 

Although a rod of iron is practically useless as a ther- 
mometer the general principle described is that on which 
most modern thermometers are constructed. 

For a thermometer to be suitable for common use, it must, 
firstly, be of such a nature that it absorbs very little heat 
from the body to which it is applied, and, secondly, its 
change in length (or volume) must be indicated in a manner 
easily visible. 

Experiment 44. 

The Making of a Mercury Thermometer. Obtain a piece 
of capillary glass tube about a foot long, and test it for uni- 
formity of bore. To do this suck a short thread of mercury 
about an inch long into the bore and measure its length 
as accurately as you are able. 

Now gently blow down the tube and force the thread of 
mercury to a new position and measure it again. Repeat 
this several times so as to bring the whole length of the tube 
under examination. 

Obviously if the bore is uniform the length of the mercury 
thread will be constant. If in any part the bore of the tube 
is increased it will be indicated by a shortening of the length 
of the thread of mercury, and, on the other hand, if the bore 
is constricted in any part it will cause the mercury thread to 
become longer. 

If the piece of tube has a uniform bore it is fit for use in the 
construction of a thermometer. If the bore is irregular, 
the tube is useless. 


Experiment 45. 

Having obtained a piece of capillary tube which has a 
satisfactorily uniform bore, hold one end of it in a blow-pipe 
flame until the end is sealed up. By cautiously blowing down 
the tube through the other end it is possible while the glass 
is still hot enough to be plastic to blow a bulb on the sealed 

The size of the bulb is decided by the type of ther- 
mometer desired. A large bulb will give a very large 
range of temperature record, but a very poor degree 
of sensitiveness. High sensitiveness and small range 
are associated with a small bulb. 

When the tube is cool enough to handle, the blow- 
pipe flame should be applied at a point about one 
inch from the open end, and the tube nearly pulled 
apart. This will produce a constricted part (as shown 
in Fig. 81) which can easily be sealed up when 

The next step is to introduce some mercury. Most 
people have experienced the difficulty of filling a 
narrow-necked bottle under a common water-tap. 
Water cannot enter the bottle without air passing 
out, and the narrow neck does not provide a passage 
for both. 

This difficulty is met with in a very aggravated 
form in the case of the thermometer tube. It is 
overcome by expelling some of the air first. 

By gently heating the bulb the contained air is 
made to expand and some of it passes out of the 
tube. If the open end of the tube is now placed in 
a bowl of mercury, as the heated air within the J^Q. gj. 
tube contracts the reduced pressure will cause a little 
mercury to flow into the tube. With perseverance a few 
drops of mercury can be forced into the bulb. 

When this is accomplished the bulb is held in the flame 
until the mercury boils. The mercury vapour produced 
replaces the air originally in the tube, and if the open end of 
the tube is now held below the surface of some mercury, the 


tube will be completely filled with mercury when the enclosed 
vapour condenses. 

The thermometer is now raised to a temperature somewhat 
in excess of the highest temperature it is desired to record, 
and a small blow-pipe flame is applied to the constricted part 
of the tube and the upper portion pulled off. The bulb and 
tube are now completely filled with mercury and hermetically 

tfhe Principle of the Mercury Thermometer. As the mercury 
cools it of course contracts, and thus leaves the upper part 
of the bore of the stem unoccupied. A very little thought 
should make it clear that, since the temperature of the 
mercury influences its volume, the temperature is indicated 
by the position of the mercury in the stem. 

This type of instrument has several merits which will 
repay a little consideration. In the first place a very small 
quantity of mercury is employed, and thus it acquires the 
temperature of the surrounding medium without the absorp- 
tion of much heat. Hence in most cases there is no appre- 
ciable fall of temperature of the body on the introduction of 
the thermometer. 

Secondly, the major part of the mercury is contained in 
the bulb f the instrument, yet any increase in volume is 
accommodated in the fine bore of the stem. A small increase 
in volume is therefore indicated by a comparatively large 
movement of the mercury. 

Again, mercury remains liquid over a very large range of 
temperatures, and thus mercury thermometers are available 
for a great variety of purposes. 

Lastly, experiment has shown that mercury expands very 
uniformly with a rising temperature. That is, equal rises 
of temperature are indicated by equal increases of volume. 

The Graduation of a Thermometer. The stems of mercury 
thermometers are marked off into equal divisions called 
" degrees," and the numbers by which they are designated 
have been determined by common practice. There are at 
the present time two scales with which the student should be 



There are certain temperatures, such as the 
melting points of common solids, and the boil- 
100 ||212 ing points of well-known liquids which are, so 
to speak, outstanding features in any tempera- 
ture scale. Of these the temperature at which 
ice melts and that at which water boils under 
normal atmospheric pressure are perhaps the 
best known. 

The Centigrade scale, as its name suggests, 
has a hundred divisions between the melting 
32 point of ice (or, what is the same thing, the 
freezing point of water), and the boiling point 
of water. The former is denoted by zero and is 
written C., and the latter is therefore 100 C. 
On the Fahrenheit scale these two tempera- 
FIG. 82. tures are respectively 32 and 212. Thus C. 
and 32 F. are equal temperatures and denote 
the temperature at which ice and water can exist together. 

Experiment 46. 

Support a large funnel in a 
stand, and having filled it with 
small lumps of ice and placed a 
vessel to catch the water as the 
ice melts, place the ungraduated 
thermometer made in Experi- 
ment 45 with its bulb well within 
the ice as shown in Fig. 83. 

Since the experiment is likely 
to be conducted in a room whose 
temperature is sufficient to cause 
the ice to melt, the thermometer 
will gradually acquire the tem- 
perature of melting ice. 

Watch the mercury carefully 
and do not mark its position 
until no further movement can 
be detected. 

This point on the stem of the 



thermometer represents the freezing point of water, and may 
be marked if a Centigrade thermometer is required, or 
32 if it is desired to have it record temperatures on the 
Fahrenheit scale. 

Experiment 47. 

Fit a flask with a cork through which two holes have 
been bored, one to carry the thermometer, and the other 
a bent glass tube as shown in Fig. 84. 

Erect the whole on a stand, and 
having placed a little water in the 
flask apply the flame of a Bunsen 
burner until the water boils. 

The thermometer and the water 
should be in such relative positions 
that the bulb of the former is at 
least an inch above the surface of 
the water. 

As the steam is formed some of 
it will condense on the bulb of the 
thermometer and gradually raise 
the temperature of the mercury up 
to that of the condensing point of 
steam (or boiling point of water). 

Wait until the mercury ceases to 
rise any further, and then mark its 
position on the stem and designate 
it 100 C. or 212 F. as the case 
may be. 

The distance on the stem be- 
tween the freezing point and boiling 

point of water may now be divided into equal divisions 
or degrees. Note that there are 100 divisions in this interval 
on the Centigrade scale and 180 divisions on the Fahrenheit 

Conversion of Temperatures. Generally speaking, the 
Centigrade scale is used for scientific purposes throughout 
the world. In this country the Fahrenheit scale is exten- 

FIG. 84. 


sively used for commercial purposes, and very largely in 
engineering work. 

It frequently happens, therefore, that a temperature on 
one scale is required on the other, and thus a ready means 
of conversion is needed. 

It is, of course, only a case of simple proportion, but the 
student should remember that it is absolutely necessary to 
work with the number of degrees above or below some fixed 
point, usually the freezing point of water. 

It will be seen that between the freezing point and boiling 
point on a Fahrenheit thermometer there are 180 degrees, 
and since these are equal to 100 degrees on the Centigrade 
scale, it follows that each Fahrenheit degree is equal to {jths 
of a Centigrade degree. 

Ex. 17. Convert 15 C. and 87 F. to corresponding read- 
ings on the other scale. 

(1) 15 C. = 15 degrees above the freezing point. 


and 1 degree C. = - degree F. 

9x15 . 
15 = g degrees F. 

= 27 

This 27 degrees F. is 27 degrees above the freezing point, 
which is 32. 

/. 15 C. = 27+32 degrees F. 
= 59 F. 

(2) 87 F. =87 32 degrees above the freezing point. 

CM -' j> )> 

And 1 degree J 1 , = - degree C. 

5x55 , 
55 = degrees C. 

This 30J degrees is above the freezing point, which is 
on the Centigrade scale. Thus 87 F.=30J C. 



A graph may be constructed for these transformations. 
Since both scales are uniform the graph is a straight line, 
and it is, therefore, only necessary to fix two points. Fig. 85 
shows such a graph in which the two points selected are the 
freezing and boiling points. 
























? 20 40 60 60 100 120 


FIG. 85. 

The student should reproduce this graph on a large scale, 
and check the conversions made in Example 17. 

The Absolute Scale of Temperature. We have already 
seen that the zero point on the Centigrade scale is the freezing 
point of water. If the thermometer is placed in ice to which 
a little common salt is added, it will be found that the mercury 
contracts below this point, showing that a mixture of ice 
and salt is colder than ice melting in the normal manner. 

Suppose such a mixture caused the mercury to fall below 
the zero point a distance equal to ten degrees on the other 


part of the scale. This temperature would be referred to 
as " minus ten degrees Centigrade " and written 10C. 

Ice mixed with some other substances will produce tem- 
peratures below that obtained by ice and salt, and certain 
mechanical means are capable of producing temperatures 
considerably lower. 

Now we shall see later that heat is a form of energy. When 
we cool a body we take energy from it. It is quite reasonable 
to suppose that energy cannot be extracted from a body 
indefinitely. There naturally comes a time when there is 
none left, and since no chemical or mechanical device can 
extract what isn't there, the body may be said to be devoid 
of all heat. 

Under these conditions the temperature is said to be 
" absolute zero." This temperature has not yet been reached, 
although some experiments have got very near to it. There 
are many reasons, however, for believing that absolute 
zero would be reached at 273 C. 

For some purposes it is convenient to construct a scale 
of temperature on which this point is denoted by 0. The 
size of the degrees is not of so much importance, but the 
centigrade degree is very suitable. 

Such a scale is called the absolute scale of temperature, 
and it is easy to see that any temperature on this scale is 
obtained by adding 273 to the reading on the Centigrade 
scale. Thus 15 C. is equal to 288 degrees absolute. This 
may be written 288 A. 

Engineers sometimes find it desirable to use Fahrenheit 
degrees on the absolute scale. Since absolute zero on the 
Fahrenheit scale is 459 F., we obtain the absolute tem- 
perature on this scale by adding 459 to the Fahrenheit read- 
ing. Thus 32 F. is equal ^to 491 on this type of absolute 

Fig. 86 shows two graphs. One gives the relation between 
temperatures on the Centigrade and Absolute scales, and 
the other indicates the relation between Fahrenheit and 
Absolute temperatures. 

These graphs are of interest because they intersect at a 
point representing 40 C. and 40 F. and 233 A. Thus 



-40 C. is equal to -40 F. The student should consider 
carefully why this is so. 

Other Thermometers. Mercury is not the only substance 
that can be used in a thermometer. For very low tem- 














FIG. 86. 

peratures mercury is unsuitable, since it freezes at 39 C. 
Thermometers for low temperatures are often filled with 
alcohol (usually coloured with a little dye in order that it 
may be easily seen). 

For extremely low temperatures, and for temperatures 


above a few hundred degrees Centigrade, thermometers of 
this type are not suitable. Various electrical and optical 
devices are employed for the determination of such tem- 

The following table is of interest and may be useful for 
reference : 

Absolute zero .... 273 C. 

Hydrogen boils .... -253 

Oxygen boils . . . . -183 

Carbon dioxide boils . . . - 78 

Mercury freezes .... - 39 

Water freezes .... 

Water boils ..... 100 

Tin melts 232 

Lead melts 327 

Mercury boils .... 357 

Zinc melts 419 

Sulphur boils . . . . 445 

Aluminium melts .... 657 

Zinc boils 918 

Silver melts .... 961 

Gold melts 1062 

Copper melts .... 1083 

Pure iron melts .... 1500 

Silica softens . . . about 1500 

Platinum melts .... 1750 

Tin boils 2270 

Tungsten melts .... 3000 

Exercises 13. 

1. Draw a line 5" long and graduate one side to represent 
the graduations of a Centigrade thermometer ; let the gradua- 
tions rise from C. to 100 C., rising by 10 C ; the gradua- 
tions on the other side to represent from 32 F. to 212 F., 
rising by 20 F. 

2. Repeat the above exercise with .the graduations on 
one side from 32 F. to 150 F., rising by 20 F., and corre- 
sponding Centigrade graduations on the other side. 


3. Draw a line 4" long and graduate one side to represent 
from F. to 150 F., graduated every 20 F. On the other 
side draw a suitable Centigrade scale to correspond. 

4. Repeat the above exercise, but graduate one side from 
10 C. to 80 C., rising by 10 C. On the other side draw a 
suitable Fahrenheit scale. 

5. Describe, with sketches, the construction of an ordinary 
merdurial thermometer. 

6. Define the terms " freezing point " and " boiling point " 
of a thermometer. 

7. Describe how you would determine the " fixed points " 
on a mercury thermometer. 

8. By means of a large-scale graph of your own construc- 
tion, convert the following temperatures : 120 C., 135 C., 
87 C., 36 C., 25 C. to Fahrenheit readings. Convert the 
following readings Fahrenheit into Centigrade : 29, 36, 
65, 85, 115, and 205. Check each result by calculation. 

9. Why are thermometer tubes usually made with a very 
fine bore ? 

10. Convert all the temperatures given in the table at the 
end of this chapter into degrees Fahrenheit. 

11. State in your own words what you understand by the 
word " temperature." 

12. A student is performing an experiment, and to take 
readings of the thermometer he lifts it out of the liquid 
whose temperature is required. Criticise his method. 

13. In 1911 Kamerlingh Onnes liquified helium and the 
temperature attained was 271*3 C. How far was he from 
the absolute zero of temperature ? 

14. The normal temperature of the human body is 98*4 F. ; 
the normal room temperature is 68 F. Convert these tem- 
peratures into degrees C. 



15. " Segar cones " (fusion thermometers) are used for 
measuring high temperatures, and the following are some 
cone numbers and their estimated softening-points : 

Estimated softening-point in degrees C. 

590 620 710 800 920 950 1090 1150 1250 
Cone numbers 

022 021 018 015 Oil 010 03 1 6 

Give these temperatures in degrees Fahrenheit and degrees 
absolute Centigrade. 

16. White and Taylor give the following colour scale : 

Name of the colour. Degrees CJ. 

Dark red heat 566 

Dark cherry red ..... 635 

Cherry, full red 746 

light red ....... 843 

Orange . . ... . . 899 

Light orange . . . . . 941 

Yellow 966 

White 1,205 

Give these values in degrees P. 

17. The following boiling points of non-metallic elements 
are given : Argon, 186 C. Chlorine, 33 '6 C. Krypton, 
-151*7 C., and Neon, 239'0 C. Express these results in 
degrees F. absolute. 

18. The following figures are taken from a steam table : 

Absolute pressure 
in Ibs. per sq. in. 

Temperature F.! ^. 

Temperature C. 




Complete the table. 


19. The critical temperature of a gas is that temperature 
above which no pressure suffices to produce a liquid. Convert 
the following table of critical temperatures in degrees C. and 
critical pressures in atmospheres into critical temperatures 
in degrees F. absolute and pressures in Ibs. per sq. in. 


Substance. '^^roSS* 


Critical pressure in 

Argon . 
Oxygen . 
Nitrogen . | 



20. In rivetting the end of the rivet is made slightly 
tapered and the hole larger than the rivet. Why is this ? 

21. Stromeyer points out that a fresh charge of fuel thrown 
on to a fire must be brought to the following temperatures 
before chemical action will take place : 

Dried peat, 435 F. Anthracite dust, 570 F. Lump coal, 
600 F., and coke 800 F. Convert these temperatures into 
degrees C. 



Linear Expansion. We have already seen in a general 
way that bodies mostly expand when their temperature is 
raised. We will now consider this matter in a little more 

In an investigation of this nature, it is always desirable 
to inquire into the simplest cases first. Hence we will 
commence with solids, and even so we will consider only their 
" linear " expansion, that is, their expansion in length. 

A great many types of apparatus are in use for the measure- 
ment of the expansion of metal rods. That shown in Fig. 87 
is a suitable form for the purpose. 

FIG. 87. 

The metal rod under investigation is generally about 
50 cms. long and has pointed ends. These fit into conical 
recesses in the metal plugs which are shown in Fig. 87 at 
the ends of a metal cylinder. 
L 161 


This containing cylinder is lagged with string or other 
material to prevent undue loss of heat. It is mounted on 
a stand so that one plug rests against a screw stop which is 
shown at the left-hand side of the figure. 

The other plug is in contact with a micrometer screw 
gauge, which enables any movement to be determined with 

The cylinder is fitted with three side tubes. Into the 

middle one a thermometer is fixed and the other two are 

used for maintaining a passage of steani through the 

In use the initial length and temperature of the metal 
rod are first determined. The rod is then put into the 
cylinder and the plugs pushed in until the pointed ends of 
the rod pass into the conical recesses provided for the pur- 

The cylinder having been placed in its support, the micro- 
meter screw is brought into contact with the right-hand plug 
and the reading taken. The micrometer screw is then 
brought back a few millimetres to allow room for expansion, 
and steam is passed through the instrument for ten minutes 
or thereabouts. 

The thermometer should now record a constant tem- 
perature not far removed from 100 C. Without interrupting 
the steam supply the micrometer screw should be brought 
into contact with the plug again and its reading taken. 

The Coefficient of Linear Expansion. It is obvious that 
the actual amount of expansion of a rod will depend upon : 

(i) The initial length of the rod. 
(ii) The rise in temperature, 
(iii) The properties of the material. 

Now it is the last of these conditions that we wish to 
investigate. Hence we generally calculate from the data 
provided by the experiment the amount of expansion which 
one may expect from a rod of unit length, when its tern- 


perature is raised one degree. This quantity is called the 
" coefficient of linear expansion." 

Ex. 18. Find the coefficient of linear expansion of brass 
from the following data : 

Initial length of brass rod . . 50 cms. 

Initial temperature of rod . . 17 C. 

Temperature recorded by thermometer 

in steam chamber .... 99 C. 

Initial reading of micrometer . . 0'78mm. 

Final reading of micrometer . . 1'55 mm. 

We see that the amount of expansion is 1*55 0'78=0'77 
mm. Converting this to centimetres we get 0*077. 

The rise of temperature was 99 17 = 82 degrees. Assuming 

that the expansion is uniform we have an expansion of ^- 
cms. for each degree. 

But the rod was 50 cms. long, and if it had only had a 
length of 1 cm. the expansion would have been only l-50th 
of that measured. 

Hence the coefficient of linear expansion of brass is 

From this example it is easy to see that : 

Measured Expansion 
Coefficient of linear expansion = . . rr ~ 

Original length x Degrees rise. 

Experiment 48. 

Determine the coefficient of expansion of a number of 
different metals. 

The results may be compared with those given in the 
following table, which will also be useful for reference. 




Degree Centigrade. 

Aluminium ..... 


Brass ...... 


Copper ...... 


Iron, Cast ..... 


Iron, Wrought .... '0000125 

Lead -0000292 

Nickel -0000128 

Platinum -0000090 

Silver -0000192 

Tin -0000223 

Zinc -0000292 

The Coefficient of Cubical Expansion. In the case just 
considered we examined only the increase in length of the rod 
resulting from a rise of temperature. It is reasonable to 
suppose that the diameter increased also. 

It occasionally happens that the increase in volume, is 
required, and then we need the coefficient of cubical expansion 
of the body. This is the increase of unit volume when the 
temperature advances one degree. 

Consider a cube whose edge is of unit length at a given 
temperature. If a is the coefficient of linear expansion of the 
material of which the cube is made, the length of the edge at 
a temperature one degree higher will be (!-}-#) 

Thus we see that : 

Volume of cube at initial temper ature = I 3 

1 higher = (l-f-a) 3 . 
But (l-fa) 3 = l+3a+3a 2 +a 3 . 
The increase in volume is therefore 3a-f3a 2 -j-a 2 . 

A reference to the table given above shows that the value 
of a is very small for all the metals mentioned. If any one 
is selected, and the value of 3a 2 -f 3 determined, it will be made 


clear that these terms have so small a value as to be quite 

Thus we may say that unit volume expands 3a units when 
the temperature is raised one degree. Hence the coefficient 
of cubical expansion is 3a. But this is three times the value 
of the coefficient of linear expansion. 

The Expansion of Liquids. Just as normal work demands a 
consideration of the linear expansion of metals, so it requires 
data referring to the cubical expansion of liquids. 

Experiment 49. 

Determine the coefficient of cubical expansion of water. 
The density bottle described in Chapter III is suitable for 
this purpose. First clean and dry the bottle and weigh it. 
Then fill it with cold water at a known temperature and 
weigh again. 

The whole is now immersed in a vessel of water in which a 
thermometer is suspended. This water is slowly raised to a 
temperature of 50 or 60 C., care being taken to keep the 
water in the vessel well stirred. In this way the water 
within the density bottle acquires the temperature of the 
outside water. 

The bottle is now removed, the outside is dried, and when 
cool it is again weighed. An example should make the 
method clear. 

Ex. 19. Find the coefficient of cubical expansion of water 
from the following data : 

Weight of empty density bottle, 9 '842 grins. 

bottle full of water at 16 C., 34'839 grins. 
,, ,, bottle and water after heating to 54C. =34*518 

As the water in the bottle was heated it expanded, and 
some of it therefore flowed out through the small channel 
in the stopper. Thus the final weighing is less by an amount 
representing the weight of the expelled water. 


Subtracting the weight of the empty bottle in each case we 
have : 

Weight of water at 16 C. = 24*997 grms. 

54 C. = 24-676 
,, ,, ,, expelled by a rise of 38 degrees =0*321 grms. 

Following lines similar to those adopted in the case of the 
metal rod, we have : 

.Coefficient of "1 Volume expelled 

cubical expansion/ "Volume remaining x degrees rise. 

Since the " volume expelled " and the " volume remain- 
ing " were at the same temperature (viz. 54 C.) these volumes 
are proportional to their weights. Hence we ma}^ write : 

Coefficient of 1 _ Weight expelled 

cubical expansion/ Weight remaining x degrees rise. 

Substituting our values we get : 

Coefficient of cubical expansion of 1 0'321 
water between 16 C. and 54 C. J24 ; 676x38 = 

It is necessary to add that water has a very irregular 
coefficient of expansion, and in the example given we have 
only determined the average coefficient over a given range of 
temperature. If another range of temperature had been 
employed a different result would have been obtained. 

The following table will show how the coefficient of expan- 
sion of water varies with the temperature. 


Coefficient of Cubical 
Expansion of Water. 

to 10 C. 


10 20 C. 


20 30 C. 


30 40 C. 


40 50 C. 


50 60 C. 


GO 70 C. 



Water is an exceptional substance in this respect, most 
other liquids being much more regular. The following table 
gives the coefficient of expansion of mercury over the same 
range of temperature. 


Coefficient of Cubical 
Expansion of Mercury. 








It will be seen that there is comparatively no variation in 
the coefficient, the actual variation being so small that it 
requires work of a very high degree of refinement to detect it. 

A discerning student may have noticed that in Experiment 
49 and Example 19, no reference was made to the fact that the 
glass vessel containing the water was also subject to the rise 
of temperature, and since the glass expanded the capacity 
of the vessel was increased. 

The coefficient of expansion of the liquid determined in 
this way is therefore too low. It is called the apparent 
coefficient of expansion. When a correction has been applied 
for the expansion of the glass, we have the " absolute " 
coefficient of expansion. The coefficients given in the tables 
above are all absolute. 

Of course the expansion of glass depends to some extent 
upon its composition, but the average coefficient of linear 
expansion of glass is about 0*000008. 

Multiplying this by 3 we get the coefficient of cubical 
expansion, and it will be observed that it is very small indeed 
compared with that of water or mercury (or indeed, other 

Hence the influence on the result is very small, and in 
addition we have the fact that in practical work liquids have 
to be held in a vessel, and it is the coefficient of apparent 
expansion which is mostly needed. 



FIG. 88. 

Experiment 50. 

Expansion of Gases. Obtain a piece of 
thin glass tube of not more than 2 mm. 
bore and about 30 or 40 cms. long. Draw 
a short thread of mercury into one end 
and seal up the other end in a Bunsen 

We now have a quantity of air enclosed 
in the tube by the mercury thread, the 
volume of the air being indicated by the 
length of the tube which it occupies. 

The tube is supported in a large beaker 
of water which also contains a thermo- 
meter as shown in Fig. 88. 

The water is slowly heated with con- 
stant stirring to keep its temperature 
uniform. The length (I) of the air column 
and the temperature of the water are 
recorded at frequent intervals. 

We will again illustrate the method by 
an example. 

Ex. 20. The following table gives corresponding readings 
of temperature and length of air column in Experiment 50. 
Find the coefficient of cubical expansion of air. 

Temperature 16 C. 
Length in cms. 24 '5 

25 C. 35 C. 45 C. 55 C. 
25-25 26-2 27 '0 28 -0 

A very casual glance at these numbers is sufficient to show 
that we are here dealing with expansion the magnitude of 
which is greatly in excess of anything hitherto experienced. 

First express the results by means of a graph, which is 
shown in Fig. 89. The points lie very close about a straight 
line, and it will be observed that the graph has been pro- 
duced backwards so as to include the temperature of C. 

If a little ice is available this point may be obtained ex- 
perimentally, and should certainly be included if means 







10 20 30 40 50 

FIG. 89. 


Selecting points on the graph we obtain : 

Volume at C. = 23-05 units. 

50 C. = 27-50 

Expansion for a rise of 50 degrees = 4*45 

The coefficient of cubical expansion of a gas is given by : 

Amount of expansion 
Volume at C. X degrees rise. 
Hence we have : 

Coefficient for air = ^Jjg; 
= 0-0038. 

Since gases expand so rapidly, it is necessary to state the 
temperature at which the volume is expressed in the above 
formula. C. has been selected for this purpose. 


The following table gives results of very careful measure- 
ments on a number of gases. 

Coefficient of cubical 

Gas. expansion between 

C. and 100 C. 




Hydrogen .... '003661 

Carbon monoxide 0*003669 

Carbon dioxide 
Nitrous oxide 


Nitric oxide .... 0*003678 

Methane .... '003690 

This table brings out one remarkable fact. The gases 
mentioned have widely different characteristics. There are 
elements and compounds, a light gas such as hydrogen and 
heavy gases like carbon dioxide and nitric oxide, yet the 
coefficient of expansion is practically the same in every case. 

It is found that some gases which are more readily liquefied 
by pressure deviate slightly from this apparently general 
rule. But sulphur dioxide, one of the worst offenders, has 
a coefficient of expansion of '00398, so the deviation is not 
very great. 

The rule is so generally folloAved that it is commonly re- 
garded as a " law " and goes under the name of Charles' law. 
It states that : If the pressure remain constant, any given 
mass of gas expands one two hundred and seventy-third of 
its volume at C. for each degree of rise in temperature. 

Note that 1/273=0 '003663, which is very close to the values 
given in the table above. 

From this it is clear that if any gas has a volume of 273 
units at C., at 1 C. its volume would be 274 and so forth. 

But these numbers represent the temperature on the 
absolute scale, and hence we may state Charles' Law in a 
much better form, thus : 


If the pressure remain constant, the volume of a given 
mass of gas varies directly as the absolute temperature. 

Since the volume of all gases is affected to so large an 
extent by variations of temperature and pressure it is neces- 
sary to state all volumes under some agreed standard con- 
ditions of temperature and pressure, and not under the 
conditions under which a gas happens to be measured. 

The standard conditions which have been agreed upon 
are a temperature of C. and a pressure corresponding to a 
column of mercury 760 mm. high. These are denoted by 
N.T.P. or normal temperature and pressure. 

Ex. 21. Some coal-gas was measured at a temperature of 
18 C. under atmospheric pressure and found .to have a 
volume of 875 c.c. The barometer stood at 752 mm. Reduce 
the measured volume to N.T.P. 

Let V = volume at C. 

and v volume at t C. 

The absolute temperature at C.=273 A. 

The absolute temperature at t C. = (273+0 A - 

Now since the volume varies directly as the absolute 
temperature we have : 

V 273 

therefore V = 



273 +t 

Again, if v 1 = volume of some gas when the pressure is p 1 
and v 2 = volume of the same gas when the pressure changes 
to p 2 by Boyle's law (see Chapter IV) we have : 

Vi Pi = 
Hence v, = 

Now if j p = 760 mm. we et : - 


Applying this to the formula for correcting for temperature 
and denoting volume at N.T.P. by V and letting v= volume 
measured at t C/and p mm. pressure, we have : 


~ 273+J X 760 

Substituting the values in the given problem we obtain : 
_ 273 X 875 752 
~ (273 + 18) X 760 
_ 273x875x752 

= 813 c.c. 

The Influence of Pressure. When we considered the 
behaviour of a gas under pressure we took precautions to 
keep the temperature constant. In the more recent investiga- 
tions of the effects of temperature change we have worked 
always under atmospheric pressure which, we have assumed, 
has not varied. 

We will now consider the effect of keeping the volume 
constant while temperature and pressure change. 

Suppose we have 273 c.c. of a gas at N.T.P. Keeping the 
pressure constant let us raise the temperature to 100 C. 
(i.e. 373 A.). The volume will now be 373 c.c. 

Now keeping the temperature constant compress the gas 
back to its original volume. Let the pressure be p. 

Boyle's law states that the product of pressure and volume 
is constant. 

Thus 273 X p = 373 X 760 


Therefore p = 760 X 5=5 

= 1040 mm. 

This is directly proportional to the absolute temperature. 
Hence we may say : 

If the volume be kept constant, the pressure of a gas is 
directly proportional to the absolute temperature. 


The Gas Equation. Let V= volume of a gas at N.T.P. 
For a given mass of any given gas this will, of course, be 
constant.) Let v=the volume of this gas when the pressure 
js p and the absolute temperature is T. 

It will be recalled that we obtained a formula for reducing 
volumes to N.T.P. in which 

273^ j>_ 
273+* 760 

Now (273+0 i s the absolute temperature at t C. and may 
be written T, 


Hence v "'- 

pv 7607 
Therefore p - = -^- 

760 V 
But F is a constant, and therefore the expression 

is a constant. It is usually denoted by the letter R. 
Thus ? = , 

or pv= .flT. 

This equation is known as the " Gas Equation." It is, 
however, only another form of the formula developed in 
Example 21. 

Exercises 14. 

1. A man is making a ring gauge and he is using J" diameter 
cast steel check pins ; their measured lengths at 62 F. 
are 8 '0000" and 8*0005", if he holds the pins in his hands until 
the average temperature rises to 72 F., calculate the new 
length of the pins. Show by sketches some simple method 
of overcoming this alteration in length. The coefficient of 
expansion of cast steel is 1-17 x 10' 5 per degree centigrade. 

2. Repeat the above example, with pins 4-9995" and 5 '0000" 
long and the same rise of temperature. 

3. A number of standard cast steel check gauges of the 
following lengths, 2", 4", 6", and 8" have been certified as 


correct at 62 F. to '00005 ". What will be the lengths on a 
hot day if the temperature of the air is 95 F, ? 

4. A cast steel cylinder is put into a furnace and heated 
from 30 C. to a temperature of 200 C. If the cylinder was 
originally 3 'GOO" diameter and 6 -0000" long, calculate its 
new length. - 

5. A 10" diameter cast-iron water pipe is at a temperature 
of 5(? F. The pipe 10 feet in length and the temperature 
falls to 32 F. What is the new length of the pipe ? 

6. A hemispherical-ended boiler of mild steel is 4 feet in 
diameter and 22 feet in length. Determine the maximum 
alteration in length if the water it contains is heated from 
62 F. to 212 F. The coefficient of expansion of mild steel 
is 1-2 x 10' 5 per degree Centigrade. 

7. A lathe is certified to have a mild steel leading screw 
whose error is not greater than -f 0*0005" per foot at 62 F. 
What alteration in length will take place when the tempera- 
ture alters to 85 F. ? 

8. In a fluid gauge there is a mild steel rule, which is cali- 
brated, and three inches represents "003" or \" per -001". In 
calibrating the instrument the following readings are taken : 

Corrections to be applied 
Nominal scale reading to the reading. 

0" -0-0000 

o-ooi* -o-oooo 

0-002" -0-00005" 

0-003" -0-0001" 

Give a list of corrections to be applied if the temperature 
of the scale is allowed to rise through 30F. whilst the object 
measured is at the standard temperature. 

9. A hard copper cylinder has to be 1*02" diameter at 62 
F. What alteration in diameter will take place if the cylinder 
is heated to 150 F. ? 

10. A mild steel gauge measured 1-9850" and a similar 
gauge for the bottom limit measured 1 -9845" when measured 
at 62 F. What alteration in length would you expect to 


find if the man using these gauges held them in his hand 
until they reached a temperature of 89 F. ? 

11. A horseshoe gauge has to measure 6 -965" inside the 
jaws and has been made accurate at 62 F. What will be 
the percentage error caused by an alteration in the tempera- 
ture of the gauge of 20 F, ? Assume the gauge is mild steel 
and only the jaws are cast steel. 

12. A tyre is to be put on a wheel and the wrought-iron 
hoop is made slightly less in internal diameter than the outside 
of the wheel. Give reasons for this and explain how the hoop 
is placed on the wheel. 

13. A compound bar consists of two strips, one of brass 
and the other of mild steel, rivetted together. The bar is 
placed edgewise in a Bunsen flame, so that both metals are 
heated equally. Give sketches, showing how the metal 
bends when heated and cooled. 

14. Why is a space left between the ends of railway lines ? 
Give sketches of the side-plates, bolts and bolt-holes which 
fasten the ends of the rails together. 

15. A railway bridge is built up of joists and is 27 feet 
long. What alteration in length will take place if the length 
is 27 feet at a temperature of 62 F. and there are then 
temperature changes ranging from 32 F. to 95 F. ? 

16. The compensated balance-wheel of a watch is made 
with the arcs of the wheel of metals of different coefficients 
of expansion. The metal with the smaller coefficient is the 
inner metal. Make a sketch and explain the action. 

17. Explain, by the aid of sketches, six uses to which the 
property of the expansion of metals with heat is put. 

18. Explain by the aid of sketches six common dis- 
advantages which occur in practical working owing to the 
expansion of metals with heat. 

19. Give reasons to show why it is advisable to have fire- 
bars in boilers loosely fitting. 

20. Describe any experiment you have performed to deter- 
mine the coefficient of expansion of metals. 


21. The pattern-maker uses a special rule called the " con- 
traction rule." ]?or cast iron this rule is " per foot longer 
than the ordinary rule. Cast iron melts at 1,500 C. Assuming 
that the coefficient of contraction is constant throughout 
the cooling range (this is not true in practice) determine 
how much a rod of cast iron one foot long at the temperature 
given will contract. Let the final temperature be 35 C. 

22. Describe in detail some method for keeping constant 
the fate of a clock. Variations in temperature ^ must not 
have any effect on the length of the pendulum. 

23. Explain what you mean when you say that '02 carbon 
steel has a mean coefficient of expansion of 11 '8 x 10~ 6 between 
the temperatures of 15 and 200 C. 

24. The distance between telephone posts is 80 yards 
and the wire used is copper. What difference in the length 
of the wire would you expect between C. and 30 C. ? 

25. Convert the table given on page 164 from a table of 
linear expansion to a table of cubical expansion. 

26. The coefficient of linear expansion of aluminium is 
'0000231. Determine the cubical expansion of a cube of 
aluminium of 3" side if the temperature of the cube is raised 
20 C. 

27. Platinite is an alloy of iron with 42% nickel. This 
alloy has the same coefficient of expansion and contraction 
at atmospheric temperature as glass. Discuss the use of 
this alloy as a wire mesh for the manufacture of armoured 
glass. What difference would you expect if ordinary wire 
netting were used instead ? 

28. In the manufacture of electric lamps a small piece 
of platinum is fused into the glass. Why is platinum used 
in preference to cheaper metals ? 

29. Invar is an alloy of iron with 36% nickel and the 
coefficient of expansion with ordinary atmospheric changes of 
temperature is smaller than that of any other metal. The linear 
coefficient of expansion ranges from 0*000000374 to '00000044 


for changes of 1 C. How does this compare with mild steel ? 
Express the comparison as a vulgar fraction in its simplest 

30. In surveying it is important that there should be as 
little alteration in length as possible in the steel tape used. 
Discuss the advantages of a steel tape 100 feet long made of 
invar as against one made of a low carbon steel. 

31. The following table gives the relative volumes of water 
at different temperatures, compared with its volume at 4 C. 

Degrees Centigrade 

4 10 20 30 40 50 70 

1-000 1-00025 1-00171 1*00425 1 "00767 1 -01180 1-02241 

If the weight of one cubic foot of water at 4 C. is equal to 
62*4245 Ibs., plot a curve of the temperature and weight of 
one cubic foot of water. 

32. An experiment was carried out on a brass rod to deter- 
mine its coefficient of expansion and the following results 
were obtained : 

Length of the rod at 18 C. . . 50 '084 cms. 
Length of the rod at 98'5 C. . . 50'159 cms. 

Determine the coefficient of linear expansion. 

33. An experiment was carried out to determine the 
average coefficient of expansion of water. A bottle weighted 
with shot was weighed in air, the bottle was then Aveighed in 
water at a given temperature and then in water at a higher 
temperature. The following are the results : 

True weight of the bottle in air . . . 82-01 gram. 
Weight of the bottle in water at a temp, of 10 C. 10'91 
Weight of the bottle in water at 25 C. . . 11 '05 

Determine the average coefficient of expansion of water 
between the temperatures given. 

34. An experiment, to determine the coefficient of expan- 


sion of air at a constant pressure, was carried out and the 
following are the results : 

Length of air column at 17 C.=24 cms. 
100 C.= 31 -8 cms. 

Determine the average coefficient of expansion of air at a 
constant pressure. 

35. An experiment was carried out by the aid of a Jolly's 
constant volume air thermometer, to find the coefficient of 
expansion of air at constant volume. The following are the 
results : 

Pressure of the atmosphere . . 752 millimetres. 
Temperature readings . . 17 -4 C. 23-8 C. 

Pressure readings in millimetres . 720 736 

Determine the coefficient. 

36. The following figures show how the volume of air 
alters with temperature in degrees Fahrenheit. Plot a graph 
showing how air expands with temperature. 

Degrees F. 

32 40 50 60 70 80 90 100 110 

Relative volumes 

1 1-021 1-043 1-066 1-089 1-110 M32 M52 1-173 

37. From the following figures plot a graph showing the 
relation between volume of air and temperature : 

Temperature in degrees F. 

32 40;TX50 60 70 80 90 

Volumes in cubic feet 
12-24 12-59 12-84 13'14 13'34 13'59 13'845 

38. From the figures given, plot a graph to show how the 
weight of air varies with pressure at a constant temperature 
of C. 

Pressure by gauge in Ibs. 

5 ' 10 20 30 40 50 60 

Weight in Ibs. per cubic foot 
0-0864 0-116 0-1455 0'204 0*263 0'3215 0'380 0*4385 


39. From the figures given, plot a graph to show how the 
weight of air varies with temperature, with a constant gauge 
pressure of 5 Ibs. per sq. in. 

Temperature <of air in degrees F. 
-20 -10 20 30 40 50 60 

Weight in Ibs. per cubic foot 
0-1205 -1184 -1455 '1395 -1366 -1388 -1310 -1283 

40. Plot a graph of barometric pressure in inches of mercury 
and altitude in feet. 

Altitude in feet 
1000 2000 3000 4000 5000 6000 7000 8000 

Barometric pressure in inches of mercurv 
30 28-88 27-8 26'76 25'76 24-79 23'86 22'97 22-11 

41. 150 cubic centimetres of air are measured at 30 C. 
If the temperature be raised to 60 C., determine the volume, 
assuming the pressure remains constant. 

42. 150 cubic centimetres of air are measured at 30 C. 
and then cooled down to 30 C. By how much will the 
volume diminish ? 

43. A long narrow tube was filled with mercury and in- 
verted in a trough. At each experiment a little of the mercury 
was run out and the volume of the air measured. The 
pressure was determined in the usual way and the following 
results are given : 

Pressure in inches of mercury 

29-2 25 21 15 10 

Volume as measured 

4-812 5-62 6-7 9'4 14 

Plot a graph of pressure and volume and determine the 
value of pv. 



Quantity of Heat. If a few cubic centimetres of water are 
placed in a test-tube and about a litre of water is put into a 
flask, on the application of a Bunsen flame to each for the 
space of one minute, a rise of temperature will result in both 

But whereas the small quantity of water is boiling, the 
larger quantity is still comparatively cold. Yet the flame 
had the same opportunity of imparting heat in both cases. 

We see therefore that a high temperature does not neces- 
sarily indicate a large amount of heat. Quantity of heat, 
while depending upon temperature differences, depends upon 
other things too. 

The Calorie. The unit of quantity of heat generally adopted 
for scientific purposes is called the " calorie." It is defined 
as the quantity of heat which is required to raise one gramme 
of water through one degree Centigrade. Strictly speaking, 
it should be from 15 C. to 16 C. 

This quantity is not materially affected by temperature, 
but in the definition it is desirable to state the interval of 
15 to 16, as the quantity of heat required to raise one 
gramme of water through one degree Centigrade is slightly 
different at other temperatures. The difference is, however, 
so slight as to be quite negligible except for very accurate 

Where very large quantities of heat have to be dealt with, 
as in engineering work, it is desirable to have a larger unit 
and the " Grand Calorie " is employed. It is equal to 1000 



The British Thermal Unit. In technical work in this 
country the " British Thermal Unit " is commonly em- 
ployed. It is the quantity of heat required to raise one pound 
of water through one degree Fahrenheit. It is often written 
as B.Th.U. 

Since the variation due to temperature is so extremely small 
it is permissible to multiply a weight of water by its tem- 
perature change to express the number of units of heat 
gained or lost. 

Experiment 51. 

The Water Equivalent of a Vessel. Obtain a thin metal 
canister with a capacity of about half a litre. Copper is a 
very suitable metal, but failing everything else a glass beaker 
may be used. Place it inside a larger vessel and pack the 
space between the two vessels with cotton wool. 

A stirrer should be made by bending a piece of stiff copper 
wire into a suitable loop, and the addition of a thermometer 
completes the equipment. 

This simple apparatus is called a calorimeter. Its use is 
best illustrated by considering an example. 

Ex. 22. The inner vessel of a calorimeter was weighed 
and 64*98 grammes was the weight recorded. About 150 c.c. 
of cold water were added and the whole now weighed 215*76 
grammes. The thermometer in the calorimeter registered 
15 C. A quantity of heated water was added (whose tem- 
perature was 82 C.) and after stirring the temperature of 
the mixture was found to be 45 C. When cool the vessel 
and water were weighed, the whole weighing 344*12 grammes. 

Subtracting the weight of the vessel from the first weighing 
we get : 

Weight of water at 15 C. = 150*78 grammes. 

The weight of water added at 82 C. = 128*36 grammes. 

Now the heat gained or lost by any quantity of water is 
the product of its weight and its temperature change. 

Hence heat gained by 

the cold water . . = 150*78 x (4515) =4,523 calories. 
Heat lost by the hot water = 128'36 X (8245) =4,749 calories. 


It will be observed that the hot water lost 226 more units of 
heat than were absorbed by the cold water, and since experi- 
ment has shown that energy in any form cannot be created 
out of nothing nor destroyed (in the sense of turning it into 
nothing), it follows that these 226 calories must be accounted 

A little thought should make it clear that whereas the 
calorimeter, the thermometer and the stirrer were initially 
at the temperature of the cold water (viz. 15 C.) at the end 
of the experiment they were at the temperature of the mixture 
(viz. 45 C.). In other words, these things had been raised 
30 degrees. It is reasonable to suppose that the 226 calories 
of heat were utilized for this purpose. 

It is a very convenient practice to estimate how much 
additional cold water would have absorbed the same amount 

of heat, under the same conditions. Obviously - = 7'5 

grammes is the required amount. 

This 7 '5 grammes is called the " water equivalent " of the 
calorimeter. It means that in all subsequent experiments 
with this calorimeter, thermometer and stirrer, the heat 
which they absorb will be the same as that which 7 *5 grammes 
of cold water would absorb. 

Specific Heat. We have already seen that the unit of heat 
is the amount of heat required to raise unit weight of water 
through unit temperature. 

Now water is an unusually difficult substance to make 
warm. That is to say, most other substances are raised 
through a given range of temperature by the absorption of 
less heat than is required to raise an equal quantity of water 
through the same range of temperature. 

The amount of heat which is required to raise unit weight 
of a substance through unit temperature is called the " specific 
heat " of that substance. We will now consider one or two 
methods by which the specific heat of a substance may be 

Ex. 23. A quantity of glycerine was placed in the calori- 
meter previously used (weight 64 '98 grammes). The whole 


then weighed 240*81 grammes. The thermometer and the 
stirrer were then placed in the glycerine, the former recording 
17 C. 

Hot water whose temperature was 76 C. was now added 
and the two liquids thoroughly mixed by stirring. The 
temperature of the mixture was 50*5 C. When cool the 
calorimeter and mixture were weighed and 373*22 grammes 

Find the specific heat of the glycerine. 

Collecting weights we have : 

Weight of the glycerine at 17 C. = 175*83 grammes. 
Weight of water at 76 C. =132*41 grammes. 

Temperature through which glycerine and calorimeter are 
raised: 50*5 17=33*5 degrees. 

Temperature through which the water falls : 76 50*5=25*5 

Heat gained by the calorimeter, etc. 

= Water equivalent X degrees rise. 
=7*5 grammes x 33*5 degrees. 
=251*2 calories. 

Heat gamed by the glycerine 

= Weight of glycerine x degrees rise x specific heat. 
= 175*83x33*5 degrees xS. 
=5890 S calories, 
where $=the specific heat of the glycerine. 

Heat lost by the water 

= weight of water x degrees fall. 
= 132*41x25*5. 
=3376*5 calories. 

Now we know that : 

Heat gained = Heat lost. 
Therefore 251*2+5890 =3376*5 
Therefore 5890 S =3376*5 - 251 *2 




Experiment 52. 

Find the specific heat of any liquid (which is not acted 
upon chemically by water) by the method illustrated in 
Example 23. 

The Specific Heat of Metals. For this purpose a known 
weight of the metal is usually heated to a known temperature 
and dropped into a calorimeter containing cold water. 

It js very necessary for accurate work to provide some 
means of heating the metal so that it may be transferred 

from the heater to the 
calorimeter without loss 
of heat. 

Fig. 90 shows a suit- 
able form of heater. It 
consists of a cylindrical 
copper vessel provided 
with another cylinder in- 

^^^^ side so that there is an 

II IM annular space through 

IM-- rJll which steam may be 

FIG. 90. 

An opening at the to 
is provided with a cor 
through which two holes 
are pierced. One of these 
carries a thermometer 
(T), and the other is 
closed with a short glass 
plug which secures a piece 
of thread on which the 
metal (M) is suspended 
so that it hangs near the 
bulb of the thermometer. 

After steam has been 
passing through such a 
vessel for ten or fifteen 
minutes, the metal will be 
raised to a temperature 


not far removed from 100 C., the exact temperature being 
recorded by the thermometer, 

The calorimeter can now be held under the lower end of 
the heater and the glass plug having been removed, the 
metal may be gently lowered into the water contained by the 
calorimeter. The following example will indicate the method. 

Ex. 24. A piece of brass weighing 157*24 grammes was 
placed in the heater just described. The calorimeter already 
employed (weighing 64 '98 grammes and of water equivalent 
7*5 grammes) was weighed with a quantity of cold water, 
the combined weight being 185*20 grammes. The ther- 
mometer being applied recorded 16*5 C. 

The metal in the heater, having reached a temperature of 
98 C., Was lowered into the water, the temperature of which 
rose to 24*5 C. 

Here we have : 157*24 grammes of brass falling through 
98-24*5=73*5 degrees. 

And 120*22 grammes of water rising 24*5 16*5 = 8 degrees. 

Now heat gained by calorimeter, etc. 
= Water equivalent X degrees rise 
=7*5x8 degrees 
=60 calories. 

Heat gained by the water 

= Weight x degrees rise. 
= 120*22x8 degrees. 
=961*8 calories. 

Heat lost by the brass 

= Weight of brass X degrees f all X specific heat. 
= 157*24x73'5x 
= 11557 8 calories. 

As before we may say that : 

Heat gained = Heat lost 
Hence 60+961*8 = 11557 8 





Experiment 53. 

Find the specific heat of a number of solids, including as 
many metals as are available. 

Students of chemistry will find it of interest to plot a grapli 
showing the relation between the specific heat of various 
metals and their atomic weights. A table should also be 
made giving the product of the atomic weight and specific 
heat 'of the metals he has experimented upon. This product 
is sometimes called the " atomic heat." 

The specific heat of all substances varies with the tem- 
perature to a greater or less extent. The following table 
gives the specific heat of a number of common substances at 
the approximate working temperature of a laboratory. It 
will be useful for reference. 


Specific Heat. 

Aluminium .... 


Copper ..... 
Gold .... 


Iron ..... 


Lead ..... 


Magnesium .... 
Mercury .... 
Nickel . 


Platinum .... 


Silver . . 




Tungsten .... 
Zinc ..... 


Glass 0*2 

Porcelain . . . . 0'26 

Silica . . . 017 

Acetic acid .... 


Ether . . . 


Ethyl alcohol 



The Specific Heat of a Gas. The general principle under- 
lying the method of determining the specific heat of a gas is 
the same as that already described in the case of solids or 
liquids. The technical details of the apparatus are, however, 
much more complex, as a gas occupies such a large volume 
compared with its weight. 

Just as in the case of other substances the specific heat of 
a gas depends to some extent on the temperature, but and 
this is not the case with other substances the pressure here 
plays an important part. 

The following table gives the specific heat of a few common 
gases at the average temperature and pressure of the atmo- 


Specific Heat. 

Carbon dioxide 
Hydrogen .... 
Nitrogen .... 
Oxygen .... 


It may be of interest to add that the specific heat of steam, 
at atmospheric pressure and a temperature of 100 C., is 0'435. 

In all these cases it is assumed that the pressure has re- 
mained constant. That is, as ,the gas is heated it expands ; 
at least, it will do so unless it is prevented by applying pres- 
sure. In that case, however, the pressure would not be constant. 

We have already seen that the atmosphere exerts a pressure 
of about 15 pounds per square inch, and if a gas expands 
under this pressure it has, so to speak, to push this pressure 
back to make room for itself. But this requires energy, and 
doubtless some of the energy of the heat has to be applied 
to this purpose. 

Hence when any gas is heated under constant pressure 
some of the heat goes to make the gas warm and some of it 
goes to supply the energy required to make room for the 



Of course this is true also of solids and liquids, but in these 
cases the expansion is so small that the energy thus absorbed 
is entirely negligible. 

From what has been said, it will be seen that if a gas is 
so enclosed that it cannot expand, that is, if its volume is 
kept constant, the specific heat will be less. 

Although it is possible directly to measure the specific 
heat at constant volume, it is often more practicable to deter- 
mine* the ratio of the specific heat at constant pressure to 
that at constant volume. 

The following table gives an indication of the quantities 
in question. 


Heat at 

Heat at 








Bromine vapour . 
Carbon dioxide 




Hydrogen . . 




The following table should also be studied : 


Ratio of Specific Heat 

at Constant Pressure to 

that at Constant Volume. 

Mercury vapour 
Helium . 

Steam . 
Carbon dioxide 
Bromine vapour 
Ethane , 
Ether vapour 












Students of chemistry will notice that the ratio is greater 
in the case of elements than in the case of compounds. 

Experiment has shown that the ratio depends on the con- 
stitution of the molecule, becoming less as the molecule 
becomes more complex. 

Isothermal Expansion. It will be recalled that when 
Boyle's law was the subject of investigation in Chapter IV 
the gas was expanded (or compressed) very slowly in order 
to maintain a constant temperature. Such a process is called 
" isothermal " expansion. For a gas to expand isothermally, 
time and facilities must be given for the heat to flow into 
the gas in order to keep the temperature up. 

On the other hand, if it is desired to compress the gas 
isothermally, time and facilities must be provided for heat to 
flow out of the gas. 

Adiabatic Expansion. If a gas is expanded (or compressed) 
with extreme rapidity there is no time for heat to flow into 
(or out of) the gas. In such cases the expanded gas becomes 
colder, or, if compression has taken place, the gas has become 

This is called " adiabatic " expansion (or compression). 
Few experimental processes are sufficiently rapid to get 
perfect adiabatic expansion, but the process is approximately 
approached in the expansion of the working fluid in the 
cylinders of some* high-speed engines. 

Exercises 15. 

1. Determine the number of calories and grand calories 
in the following cases : 

1. 20 Ibs. water, Rise in temp, in degrees Centigrade 30 

2. 56 Ibs. water 54 

3. 37 -5 Ibs. water ,. 85 

4. 42-5 Ibs. water 100. 

2. In the above examples determine the number of B.Th.U. 
of heat given to the water. 


3. Define the " Grand Calorie," the " Calorie," and the 
" British Thermal Unit." 

4. Why is water the best substance to use in a hot-water 
bottle ? Contrast the use of water with that of powdered 
silica and any other substance you think suitable. 

5. The following results refer to a Diesel engine. The 
weight of cooling water in Ibs. per brake horse-power used 
per hour and the rise in temperature in degrees Fahrenheit 
are given. 

Determine the B.Th.U. per brake horse power per hour 
carried away by the cooling water. 

Cooling water used in Ibs. per B.H.P. 145 18'6 45 

Rise in temp. . 29'2F. 118F. 55'3F. 

6. In an engine test the total cooling water used in Ibs. 
per hour was 115 '65, the inlet temperature of the water was 
54 F. and the outlet temperature was 103 F. Determine 
the B.Th.U. of heat carried away by the cooling water per 

7. The heat supply of an engine in B.Th.U. per brake 
horse power per hour was, (a) 14200, (6) 12000, (c) 11000, (d) 
10200. What are these results in B.Th.U. per B.H.P. per 
second. Give these results in calories per second. 

8. Professor Robinson gives the following calorific values. 

Calorific value in 
Oil. B.Th.U. per Ib. 

Refined Royal Daylight . 20286. 

Refined petroleum .... 19885 

Double refined 19955 

Pratt's motor spirit .... 18610 

Give these calorific values in calories per gramme. 

9. Judge, " Handbook for Modern Aeronautics," gives 
the estimated fuel consumption in pints per hour for the 
following types of engines : 

Curtis X 5 . . 76 pints per hour at 6000 feet. 

ABC Dragon Fly . 256 

B.R.I . . 129 

B.R.2 , 196-8 


Assuming that petrol of 0'68 specific gravity and having a 
heat value of 19200 B.Th.U. per Ib. was used; determine 
the B.Th.U. used per hour. 

10. Determine the water equivalent of 0*75 Ibs. of each 
of the following metals : Aluminium, copper, iron, lead, 
magnesium, tin, and zinc. 

11. Explain what is meant by " specific heat." 

12. A gas ring weighs 3 pounds and is at 62 F. After 
boiling water its temperature is 150 F. What is the water 
equivalent of the ring and how many B.Th.U. has it absorbed ? 

13. In an experiment to determine the mechanical equiva- 
lent of heat the following results were obtained : 

Water in the drum, 300 grammes. 

Water equivalent of the drum and thermometer, 40*75 grms. 
Original temperature of the water, 15*92 C. 
Final temperature of the water, 16*95 C. 
Determine the calories of heat generated. 

14. An experiment was carried out to determine the 
specific heat of iron tacks. The following are the results : 

Weight of the copper calorimeter, 67*135 grammes. 

Weight of calorimeter and water, 101*43 grammes. 

Original temperature of the water, 13 C. 

Temperature of the tacks before dropping into the water, 
97 C. 

Temperature of the water and tacks after mixing, 16*2 C. 

Weight of the calorimeter, water and tacks, 111 *75 grammes. 
Determine the specific heat of the tacks. 

15. An experiment was carried out to determine the water 
equivalent of a copper calorimeter. 

Weight of the calorimeter, 107*65 grammes. 

Initial temperature of the calorimeter, 13 C. 

Temperature of warm water before placing in the calori- 
meter, 41*3 C. 

j* Final temperature of water and calorimeter, 39 C. 
M Weight of the calorimeter and the water, 236*22 grammes. 
Determine the water equivalent from these results and 
also by assuming the specific heat of copper. 


16. An experiment was carried out to determine the 
specific heat of copper and the results are as follows : 

Weight of the copper cylinder in grammes, 93 '71. 
Weight of the inner cylinder or calorimeter, 107 '65 grammes. 
Weight of the calorimeter plus added water, 292:5 grammes. 
Initial temperature of the water in the calorimeter, 12 '5 C. 
Temperature of the copper cylinder, 99*7 C. 
Final temperature of the water and the copper cylinder, 
16-5 C. 

Determine the specific heat of the copper cylinder. 

17. To take into account the variation of specific heat with 
temperature, the specific heat of copper may be written as 
follows : 

Specific heat of copper at t C.,=0-0917+0-000048t. 
Calculate the specific heat of copper for the following 
temperatures : 50, 100, 150, 200, 250, and 300 C. Plot a 
graph of temperature and specific heat. 

18. A glass vessel at 30 C. weighs 980 grammes. If 2000 
grammes of water at 40 C. are poured into the glass vessel, 
determine the final temperature of the water. Take the 
specific heat of the glass as 0'12. 

19. Determine the B.Th.U. of heat carried away per hour 
by the cooling water in a 50 I.H.P. gas engine. Use the 
following formula : 


w t t l = 0-30 X I.H.P. X ~ X 60. 

Where w=lbs. of water per hour. 

t= final temperature Fahrenheit. 
t l = initial temperature Fahrenheit. 
I.H.P. = indicated horse power. 

20. In a gas engine the heat distribution is as follows : 

Work as indicated horse power . . 27*1% 

Carried away by the jacket water . . 49*5% 
Carried away by the exhaust gases . . 23 '4% 

Total 100% 


If the heating value of the fuel is 1041 B.Th.U. per cubic 
foot, show how the heat in each cubic foot of gas is dis- 

21. Determine the thermal capacity per degree Centigrade 
of a line of hot water pipe, 3" internal diameter, 4" external 
diameter, 30 yards long and made of cast iron. Neglect 
the flanges. 

22. The following results for an economizer are given : 

Temperature of the water as Temperature of the water as 

it enters the economizer. it leaves the economizer. 

84-2 F. 196-2 F. 

40 F. 185-4 F. 

101 F. 237 -0F. 

Determine how many B.Th.U. per 100 Ibs. of water are 
given up as it passes through the economizer. 

23. The following table shows pressures and volumes in 
a single stage compressor. Column 2 gives volumes with 
constant temperature or isothermal compression, column 3 
volumes as obtained in actual compressors, whilst column 4 
gives volumes for adiabatic compression. Plot three curves 
showing pressure as abscissae and the three different volumes 
as ordinates. Also prove that pv in every case of iso- 
thermal compression gives a constant : 

Pressure Volume at 

absolute. constant temp. Volume. Volume. 

14-7 1 1 1 

15-7 0-9363 0-948 0'954 

16-7 0-8803 0-903 0'910 

17-7 0-8305 0-862 0'876 

18-7 0-7862 0-825 0-841 

24-7 0-5952 0'660 0692 

29-7 0-4950 0'570 0'607 

34-7 0-4237 0'503 0'544 

24. If the total volume of a cylinder containing air at an 
absolute pressure of 14*7 Ibs. per square inch is 12 cubic feet, 
what will the pressure become when the piston has moved 



so as to reduce this volume to 5 cubic feet, the compression 
being isothermal ? 

25. With isothermal compression pv=C. Plot graphs 
of pressure and volume for the following gases : air, oxygen 
and nitrogen. Let the gas be at 14 "7 Ibs. per sq. in. pressure 
and take pressure increasing by 5 Ibs. to 34*7. 

Atmospheric air at 14 '7 Ibs. per sq. in. and 62 F. occupies 
a volume of 13 '14 cubic feet for 1 Ib. 

Oxygen as above 11*88 and nitrogen 13*54. 



Experiment 54. 

Change of State. Arrange the apparatus shown in Fig. 84 
so that the flask is about half full of cold water, and the 
thermometer has its bulb just below the surface of the 

Place a small flame of a Bunsen burner beneath the flask, 
and read the thermometer at regular intervals, say every 
half minute. 

Commencing with the water cold, continue the readings for 
a few minutes after the water has boiled. 

Plot a graph showing the relation between the time and the 

Now a consideration of this graph will show us that at 
first the rate at which the water gains heat is fairly uniform, 
and if the weight of the water is known it is possible to state 
the number of calories per minute which are absorbed by the 

Near the boiling point, however, the rate of absorption 
appears to fall off, and when the water actually begins to 
boil the temperature remains constant. 

Now it is unreasonable to suppose that the burner ceased 
to supply heat at the moment when the water boiled. Yet 
if the heat is supplied we must decide on its destination. 

Latent Heat. The behaviour of the water is explained thus : 
At first the heat supplied to the water was employed in 
increasing its temperature. This heat is called " sensible 
heat," because it is heat which is recognized by our sense 
of touch. 

When the water commenced to boil the heat which passed 



into it was used to supply the energy required to change the 
state of the water from the liquid to the gaseous state. This 
heat is called " latent heat." 

Let us think for a moment how events would be affected 
if no latent heat were required. Suppose a pint of cold water 
were put into a kettle and the latter set upon a fire. The 
water would gradually get hot and ultimately the boiling 
point would be reached. At this moment the whole of the 
water would instantaneously pass into steam, which would 
occupy a space of approximately 1,700 pints. In other 
words, a very violent explosion would occur. 

Fortunately this is not the case. When the water reaches 
the boiling point it passes into steam at a rate which is deter- 
mined by the supply of the necessary latent heat. 

The number of units of heat which are required to convert 
unit weight of water at the boiling point into steam at the 
same temperature is called the " latent heat of evapora- 
tion of water." Very often it is referred to as the " latent 
heat of steam." 

The method of determining this quantity is again best 
illustrated by taking an example. 

Ex. 25. Using the calorimeter, etc., of previous 
experiments (weight 64*98 grms. and water equiv- 
alent 7*5 grms.) a quantity of cold water is 
introduced. It now weighs 270*10 grms. The 
temperature of this water is 15*5 C. 

Steam is now passed into the water by means of 
a glass tube which emerges from a " steam trap," 
which is shown in Fig. 91. This simple appliance 
serves to remove the particles of water formed by 
the steam condensing during its passage from the 
boiler to the calorimeter, and thus only " dry " 
steam (as it is called) enters the calorimeter. 

When the passage of steam was stopped the 
thermometer in the calorimeter recorded 20 C. 
On cooling, the calorimeter and water weighed 
271*74 grms., the increase in weight representing 
FIG. 91. the weight of steam which had been condensed. 



Collecting the data we have : 

Weight of water at 15 C.=205'12 grms. 

Weight of steam condensed = ! 64 grms. 

Rise in temperature of water and calorimeter 20J 15 J = 
4J degrees. 

Fall in temperature of condensed steam 100 20J = 
79| degrees. 

Now heat gained by calorimeter, etc. = Water equivalent X 
degrees rise. 

7 '5 grms. x4J degrees. 
=35 -6 calories. 

Heat gained by water : 

= weight of water X degrees rise. 
=205 '12 grms.x4| degrees. 
=974-3 calories. 

Heat lost by steam in condensing to water at 100 C. : 
= 1-64 grrns.x.L. 
= 1*64L calories. 
where L is the latent heat. 

Heat lost by this condensed steam in falling from 100 C. 
to 20i C. : 

= 1-64 grins. x79f degrees. 
= 130 '5 calories. 

Equating heat gained and heat lost we have : 

And jL = 536 calories per gramme. 

Latent heats may be expressed in calories per gramme or 
B.Th.U.'s per Ib. Since the gramme and the Ib. enter into 
the unit of heat it is only the temperature scale which affects 
latent heats. That is, a latent heat in the latter units is 
? of that in the former units. 

The best results give the latent heat of vaporisation of 
water as 537 calories per gramme or 967 B.Th.U.'s per Ib. 
The latent heat of vaporisation of ether is 91 '3 and of ethyl 
alcohol 206 calories per gramme. These values all refer to 
vaporisation at atmospheric pressure. 


Experiment 55. 

Find the latent heat of evaporation of water by the method 
described in Example 25. 

Latent Heat of Fusion. Just as it needs heat to convert 
water into steam after it has been raised to the boiling point, 
so it needs heat to melt a solid after it has been raised to the 
melting point. The number of units of heat which are 
necessary to convert unit weight of a solid into a liquid at 
the melting point is called the " latent heat of fusion " of 
that solid. 

Ex. 26. The calorimeter, weighing 64 '98 grms. (of water 
equivalent 7 '5) had some water placed in it and its weight 
recorded. The combined weight was 192*39 grms., the 
temperature being 18 C. 

A small lump of ice was carefully dried by means of blotting- 
paper and dropped into the calorimeter. The water was 
stirred until the ice melted, and its temperature was then 
12 C. The weight was now 201 '8 grms. 

To find the latent heat of fusion of ice, first collect data 
as follows : 

Weight of water at 18 C.= 127*41 grms. 
Weight of ice =9*41 grms. 

Rise of temperature of melted ice = 12 degrees. 
Fall of temperature of water = (18 12) = 6 degrees. 

Heat gained by ice in melting and thus changing from ice 
at C. to water at C. : 

= Weight of ice X latent heat. 
= 9'41 grms.x L. 
=9*41L calories. 

Heat gained by melted ice in rising from C. to 12 C. : 

= Weight of ice x degrees rise. 
=9'41 grms.X 6 degrees. 
=56 '5 calories. 


Heat lost by calorimeter, etc. : 
= Water equivalent X degrees fall. 
= 7'5 grms.xG degrees. 
=45 calories. 

Heat lost by water : 

= Weight of water X degrees fall. 
= 127 '41 grms. X 6 [degrees. 
=764 '5 calories. 

Now heat gained = heat lost. 
.-. 9-41Z,+56-5=45+764-5. 

/. L=80 calories per gramme. 

The best experiments give a value for the latent heat of 
fusion of ice of 79 '7 calories per gramme, which is equivalent 
to 143-3 B.Th.U. per Ib. 

Experiment 56. 

Find the latent heat of fusion of ice by the method described 
in Example 26. 

In cold weather it may be desirable slightly to heat the 
water in the calorimeter. It should be done after weighing, 
and the temperature to which it is raised should not be 
over 20 C. 

It will be observed that the temperature of the ice is 
assumed to be C. This will be so unless the room in which 
the experiment is conducted is at a temperature well below 
the freezing point. This is not likely to be the case. 

Any solid which can be melted has, of course, a latent heat 
of fusion. The following table gives the melting point and 
the latent heat of fusion of a number of common metals. 

The method adopted in the determination of these latent 
heats varies with the nature of the metal. In no case, 
however, does the determination form a suitable experiment 
for the student at this stage of his work. 




Melting Point, i Latent Heat of 
Degrees Fusion in Calories 
Centigrade. per Gramme. 

Aluminium . 












t Iron .... 



Lead .... 






Nickel .... 






Silver .... 



Tin .... 



Zinc .... 



Change of Volume on Melting. Most solids when they melt 
increase in volume. Ice is an exception to this rule. Indeed, 
water is an abnormal substance altogether in this respect. 

Consider a quantity of water at, say, 15 C. If this is 
cooled it contracts until a temperature of 4 C. is reached, 
when water reaches its maximum density. From 4 C. to C. 
the water expands as it cools. On turning into ice a con- 
siderable expansion occurs, 1 c.c. of water at C. becoming 
1-09 c.c. of ice at C. 

One of the effects of this abnormal behaviour is the occur- 
rence of burst water-pipes in frosty weather. As the water 
within the pipe turns into ice it expands to the extent of 9 
per cent, of its volume. 

This expansion is absolutely necessary if the ice is to be 
formed, and a gigantic pressure is necessary to keep water 
liquid below C. 

Ordinary waters-pipes cannot withstand this pressure, and 
consequently burst. Very often the fact is not made evident 
at once because the water has now become solid ice. When 
the temperature rises, however, the ice melts and a serious 
leakage indicates the burst pipe. 



Just as great pressure applied to water will prevent its 
freezing, so a similar pressure applied to ice will melt it. 

If a large block of ice be supported on two trestles as shown 
in Fig. 92, a steel wire may be passed over it and the free 
ends joined underneath so as to support a heavy weight. 
(A half hundred-weight forms a suitable load.) 

The pressure of the wire on the upper surface of the ice 
is usually sufficient to melt the ice immediately beneath it, 
and the wire of course sinks into the narrow channel of 
water which it produces. 



FIG. 92. 

But ice cannot melt without the absorption of its latent 
heat, and since this is not forthcoming from external sources 
it has to supply the heat itself. Hence the water which is 
produced is at a temperature well below the freezing point. 

Since this water is now above the wire it is no longer 
subjected to pressure, and it therefore immediately returns 
to the solid state, that is, ice. 

The process is continued, and in time the wire will pass 
entirely through the block of ice without cutting it into 
two pieces. 

Freezing Water by Boiling. We have already seen that the 
boiling point of water depends upon the pressure to which it 



is subjected. The following tables show the boiling point 
of water at various pressures, commencing with the normal 
atmospheric pressure in each case. 

Pressures below one Atmosphere. 

Boiling Point 
of Water. 

In mm. of mercury. 

In Ibs. per sq. inch. 



100 C. 



90 C. 



80 C. 



70 C. 



60 C. 



50 C. 



40 C. 



30 C. 



20 C. 



10 C. 




Pressures above one Atmosphere. 

Boiling Point 

of Water. 

In mm. of mercury. 

In Ibs. per sq. inch. 



100 C. 






120 C. 



130 C. 



140 C. 



150 C. 



160 C. 



170 C. 



180 C. 



190 C. 



200 C. 



FIG. 93. 

Experiment 57. 

Obtain a spherical flask and fit 
it with a rubber stopper. Half 
fill the flask with water and boil it 
for a few minutes. After a time 
the air which occupied the space in 
the flask above the water will have 
been expelled and replaced by 

Now remove the flame and insert 
the stopper. Invert the flask and 
support it in a stand as shown in 
Fig. 93. 

By this time the water will have 
cooled somewhat below 100 C., 
yet if cold water be poured over 
the flask, the water inside the flask 
will boil vigorously. 

This is due to the cold water condensing the steam which 
occupies the space above the water. The condensation, of 
course, reduces the pressure, and although the water was not 
hot enough to boil under the higher pressure, it is able to 
do so when the pressure is reduced. 

Experiment 58. 

If an air-pump, such as that shown in Fig. 21, is available, 
it may be used for the purpose of reducing the pressure 
instead of the method just described. 

Support a thermometer in a vessel of warm water and 
place the whole under the bell- jar of an air-pump. 

As the pump is operated the pressure falls and the water 
ultimately commences to boil. The boiling will continue 
if the pressure is kept sufficiently low. 

The thermometer will show, however, that as the water 
boils its temperature falls. 

The reason is similar to that which caused the ice to melt 
under pressure and to revert to ice when the pressure was 

In this case we have water being made to boil without 


supplying it with the necessary latent heat. Hence every 
gramme of water which is converted into steam under these 
conditions has to take over 500 calories of heat from the 
remaining water. 

This, of course, causes a considerable fall of temperature. 
In addition it makes it necessary to reduce the pressure 
still further if the boiling is to continue. 

It will readily be seen that if the pump were approximately 
perfect it would be possible to reduce the temperature to 
such' an extent that the remaining water would begin to 

Another Effect of Latent Heat. Most bathers have ex- 
perienced conditions when it has felt warmer in the water 
than in the air. This is often the case when a wind is blow- 
ing. Yet a thermometer frequently shows that the reverse 
is actually 'the case. 

When the surface of one's body is beneath the water, it 
" feels " the temperature of the water. When it is exposed 
to the air, however, the film of water commences to evaporate 
(especially if there is a " dry " wind), and the latent heat 
which is necessary for this vaporisation is taken from one's 
body and produces the sensation of cold. 

It will be seen that the actual temperature of the air has 
very little influence on this sensation. 

In certain hot countries (e.g. Portugal) coarse earthen- 
ware jars are used to keep drinking-water cool. These jars 
are very porous, and when filled with water some of it soaks 
through to the outside. 

This film of water readily evaporates in the dry air, and the 
latent heat (or at any rate a large proportion of it) is absorbed 
from the water within the jar. Thus the latter is kept 

It has been said that it is not " hot enough " in this country 
to use this method of cooling water. It would probably be 
more correct to say that the air is not often sufficiently 


Exercises 16. 

1. Describe in detail a method of determining the latent 
heat of steam. Point out the principal sources of error in 
the experiment and the precautions to be taken to minimize 

2. In the above experiment state how the results are 
affected by using steam that is not " dry." 

3. Suppose that you have a small quantity of ice at the 
melting temperature, and that you gradually melt it. What 
changes of temperature and volume does the ice undergo ? 

4. Describe fully how you determine the melting point of 
paraffin. What are the changes that you observe during 
the change of state ? 

5. State how you would determine the latent heat of 

6. An experiment was carried out to determine the latent 
heat of steam and the following were the results obtained : 

Weight of copper calorimeter in grammes, 82*5. 

Weight of water used for condensing the steam, 98*4 

Initial temperature of the above water, 17 '3 C. 

Maximum temperature of the water after condensing the 
steam 58 C. 

Weight of the condensed steam, 7*4 grammes. 

Determine the latent heat of steam in calories per gramme 
and B.Th.U. per Ib. 

7. An experiment was carried out to determine the latent 
heat of ice and the following are the results : 

Weight of the copper calorimeter, 105 '87 grammes. 
Weight of the calorimeter and the water, 274 '7 grammes. 
Original temperature of the water in the calorimeter, 30 C. 
Final temperature of the water after mixing well with the 
dry ice, 20 C. 

Weight of the water, ice, and calorimeter, 292*7 grammes. 
Determine the latent heat of ice in calories per gramme. 


8. The results of an experiment are given which was 
carried out to determine the latent heat of steam : 

Weight of calorimeter, 107'1 grammes. 

Weight of the thermometer, 128*35 grammes. 

Weight of calorimeter, thermometer, and water 323 '9 

Temperature of the water in the calorimeter, 16 % 7 C. 

Final temperature of the water after condensing the 
steam, 41 C. 

Weight of the water, calorimeter, thermometer, and the 
steam, 332*4 grammes. 

Determine the latent heat in calories per gramme. 

9. The latent heat of steam varies with the pressure, and 
may be calculated from the following formula : 

Latent heat = 1114 0*7^ 

where t is the temperature in degrees Fahrenheit of the 
boiling point of water at the given pressure. Calculate 
the latent heat for the following temperatures : 212, 222, 
232, 242, 252, and 262 F. Plot a graph of latent heat and 
temperature. What are the units in which the latent heat 
is expressed ? 

10. The total heat of steam includes the sensible and 
the latent heat and may be calculated from the formula : 

H (total heat) = 1082 + 0'305 t F. 

Calculate the total heat of steam for the following tem- 
peratures : 212, 220, 230, 240, 250 and 260 F. Plot a graph 
of H and t. 

11. Watt, the great inventor and engineer, has given a 
full account of the early experiments he carried out which 
led up to his first patent. Steam was carried from a kettle 
by means of a piece of glass tubing to a cylindrical glass jar 
containing water. At the completion of the test the water 
in the glass vessel was then found to have increased about 
J part from the condensed steam. Consequently, water 
converted into steam can heat about 6 times its own weight 
of well water to 212, or till it can condense no more steam. 


Watt had to ask Dr. Black to explain this. Give your 

12. In a lecture by Geo. Babcock at Cornell University 
on " The Theory of Steam Making," the following occurs : 
" It follows that if we could reduce steam at atmospheric 
pressure to water, without loss of heat, the heat stored within 
it would cause the water to be red hot ; and if we could, 
further, change it to a solid, like ice, without loss of heat, 
the solid would be white-hot, or hotter than melted steel, 
it being assumed that the specific heat of water remained 
normal/' Explain this. 

13. The following also occurs in the same lecture : " The 
heat which has been absorbed by one pound of water to 
convert it into one pound of steam at atmospheric pressure 
is sufficient to have melted 3 pounds of steel or 13 pounds 
of gold/' Taking the necessary figures from the tables given 
prove the truth of this statement. 

14. If some volatile liquid is placed on the hand, the hand 
feels cold. Explain this. 

15. A person has a headache and places a handkerchief 
with eau de Cologne on it across his forehead. Give reasons 
for this. 

16. Experiments were carried out to determine the 
efficiency of small oil furnaces (see article by the authors, 
English Mechanic, June, 1912), and the following are some 
results : 

Weight of metal Weight of petroleum 

Material. melted. used. 

Cast iron 5 Ibs. 4 Ibs. 

Aluminium 2 1*2 

Lead 4 0*416 

Taking the petroleum as having a calorific value of 18,500 
B.Th.U per lb., determine the B.Th.U. used in melting the 
metal. Determine the B.Th.U. required to melt each batch of 
metal. If the thermal efficiency is the B.Th.U. required to 
melt the metal divided by the B.Th.U. used, determine the 
thermal efficiency in each case. 


17. A test was carried out in a cupola and 1,232 Ibs. of 
coke were required to melt 20,160 Ibs. of cast-iron. Deter- 
mine the total heat required to melt this quantity of metal, 
the total heat in the coke and the thermal efficiency. The 
calorific value of coke is 13,000 B.Th.U. per Ib. 

18. The table here given shows how the sensible and latent 
heats of steam vary with the pressure : 

Pressure in Ibs. per sq. in. Temperature F. Latent heat. 

1 101-99 1043-0 

2 126-27 1026-1 
5 162-34 1000-8 

10 193-25 979-0 

15 213-03 965-1 

20 227-95 954-6 

30 250-27 938-9 

40 267-13 927 

Plot a graph of pressure and temperature of boiling 
(column 2) and a graph of pressure and latent heat. 

19. What is meant by the " boiling point " of a liquid ? 
How is it affected by change of pressure ? 

20. Describe any form of boiler you have seen and state 
how the pressure is kept from rising to a dangerous 

21. Give a sketch of a domestic hot- water supply system 
and explain how the pressure is prevented from rising. 

22. The following statement is given in a catalogue : 
" Well designed boilers under successful operation will 
evaporate from 7 to 10 pounds of water per pound of first- 
class coal." 

Suppose that the boiler is working at 150 Ibs. per sq. in. 
and the total heat given to the steam per Ib. of water is 1,191 '2 
B.Th.U., whilst the calorific value of the coal is 13,800 B.Th.U., 
determine the thermal efficiency for the two cases given. 

23. Ten pounds of water are enclosed in a small steam 
boiler, the temperature of the water is 68 F. How much 


heat must be given to the water to generate 8 pounds of 
steam at atmospheric pressure ? 

24. Twelve pounds of steam at 100 C. are blown into a 
tank containing 300 Ibs. of water at 40 C. Find the resulting 



Conduction. We have frequently mentioned the fact that 
heat " flows " from one body to another according to tem- 
perature differences. In this chapter we will consider the 
methods by which heat can pass from one body to another. 

Experiment 59. 

Obtain a few pieces of thick wire of different metals, but 
of the same dimensions, about 5 or 6 inches long. Holding 
each specimen by one end, apply the other end to the Bunsen 

It will be observed that the heat absorbed from the flame 
travels along the wire and ultimately reaches the fingers. 

The student should find no difficulty in discerning between 
" good conductors " like copper and those metals which 
convey heat less readily. 

The property which all forms of matter possess in a greater 
or less degree, which enables the individual particles to pass 
heat from one to another, is known as " conduction." 

Of course, there are good conductors and bad conductors. 
Generally speaking, all metals are good conductors, although 
some are better than others ; gases, on the other hand, are 
very bad conductors of heat. 

For purposes of comparison we use a quantity which is 
called the " thermal conductivity " of a substance. It is 
the number of units of heat which, in unit time, will pass 
across a cube of unit edge, when the opposite faces are at 
temperatures differing by one degree. 

Thus, in the C.G.S. system it is the number of calories of 
heat which will pass in one second across a cube of one centj- 




metre edge when the opposite faces are at temperatures 
differing by 1 C. 

The following table gives the thermal conductivity in 
C.G.S. units for a number of substances. 


Thermal Conductivity. 



Copper .... 




Iron .... 


Lead . . 




Mercury .... 


Nickel .... 


Platinum .... 






Zinc .... 


Cement .... 


Cotton Wool 


Ebonite .... 




Glass .... 


Water .... 




Hydrogen .... 


Oxygen .... 


It will be observed that, compared with the metals, even 
the worst of them, water is a very bad conductor of heat. 

Experiment 60. 

Load a small piece of ice with a strip of sheet lead so that 
it will sink into water. Drop it into a test-tube nearly full 
of water, 



FIG. 94. 

Having supported the test-tube in a 
slanting position, apply a small flame of a 
Bunsen burner near the top of the tube, 
just below the surface of the water, as 
shown in Fig. 94. 

Heat can only reach the ice by conduc- 
tion through the water or the glass, and 
since the latter is very thin the majority of 
the conduction takes place through the 
water. It will be found possible to boil 
the "water at the top of the tube for some 
considerable time before the ice melts, thus 
showing water to be a very poor conductor. 

Experiment 61. 

Convection Currents. Support a beaker, half full of water, 

on a tripod with the small flame of a Bunsen burner beneath 

it. A few fragments of some light material should be put 

into the water ; tiny pieces of red blotting-paper are very 


As the water is heated it will be observed that there is an 

upward current of water immediately above the flame, 

while near the outer edge there is a downward current. 

These movements of the water will be rendered visible by 

the motion of the particles of blot- 
ting-paper which have been put into 

the water. Fig. 95 shows the 

general arrangement. 

If a thermometer is supported 

with its bulb at the point marked 

A, while another thermometer has 

its bulb at B, a considerable differ- 
ence of temperature will be noticed. 
It will be seen that the heat of 

the flame passes through the glass 

by conduction and heats the film 

of water in contact with the glass. 

Expansion necessarily takes place 

which renders the warm water lighter 

than the surrounding water. FIG. 95. 



Hence the warm water rises in a constant stream up the 
middle. The colder water at the edges has no option but to 
flow in and take its place. 

This in turn gets heated and rises, and so the circulation 
continues. These currents are known as " convection 

It will be observed that in the case of conduction of heat, 
the heat is passed on from particle to particle. With con- 
vection, however, a particle absorbs a certain amount of 
heat and then moves off, carrying the heat with it. 

Domestic Hot Water Systems. In many kitchen ranges 
there is fitted a small iron boiler (B, Fig. 96), between the 
grate and the flue. One pipe passes from the top of this to 
the storage tank (C) in the upper part of the house. The 
tank is generally con- 
nected with a constant 
level cistern. 

As the water in the 
small boiler becomes 
hot it ows up one 
pipe (shown black) to' 
the storage tank (7, ' 
while cold water flows 
from the latter, down 
the other pipe, into 
the boiler. 

Here we have circu- 
lation maintained by 
means of convection 
currents. Taps are fit- 
ted, where needed, on 
the " hot " pipe, that 
is, the one up which 
the hot water flows. 

A general arrange- 
ment of such a system 
is shown in Fig. 96. 
Of course, convection FIG. 96. 


currents can be produced in any fluid. Winds are largely con- 
vection currents in the atmosphere and many forms of 
ventilation are an application of this phenomenon. 

Formation of Ice on Water. It will be convenient here to 
revert to the effects of the abnormal behaviour of water 
under change of temperature. Consider a river in frosty 
weather. As the water cools it contracts, and because it is 
specifically heavier than the water beneath it, it sinks to 
the Bottom and the warmer water rises to the surface, and is 
cooled in its turn. 

After a time the whole of the water is at a temperature of 
4 C. Now it will be recalled that at this temperature 
water attains its maximum density. Hence, as the surface 
water is cooled still further it expands, and this, making it 
specifically lighter, causes it to remain on the surface. 

After a time the uppermost layer of water has been cooled 
to C., and on the withdrawal of more heat a film of ice is 

As soon as the surface of the water is covered with a film 
of ice, no more ice can be formed except by the withdrawal 
of the latent heat by conduction through the ice already 

The thermal conductivity of ice is 0'0022, and hence we 
see why it is that a prolonged frost is necessary to produce 
thick ice. 

If water behaved in a manner similar to that of most other 
liquids, that is, if it contracted as the temperature fell right 
down to freezing point and then contracted still further on 
solidification, ice would be specifically heavier than water. 

This would cause ice to form at the bottom of the rivers 
first and solidification would then take place upwards. 

If such conditions prevailed a quite ordinary winter would 
be sufficient to freeze most rivers solid. Experiment 60 will 
show with what difficulty the ice would be melted by the 
application of heat from above. 


Experiment 62. 

Half fill a test-tube with paraffin wax cut up in pieces about 
the size of a pea. 

Very slowly melt the wax by the application of heat. 

Note that the solid material is heavier than the liquid, 
since it sinks in the liquid. 

Now allow the wax to cool until it is solid again. Note 
how the surface shrinks down as the solidification proceeds, 
showing that a marked contraction occurs on passing from 
the liquid to the solid state. 

Articles made of cast iron are always more or less rough 
owing to the sand used, but to prevent the molten iron 
from shrinking away from the mould, all large castings are 
" fed." The slowly-cooling iron is forced into the mould by 
an iron rod worked up and down. 

The pattern-maker has to make the pattern somewhat 
larger than the required size of the casting because the liquid 
iron contracts on solidification. 

Radiation. So far as we have considered the question, 
whether heat is transferred by conduction or by convection 
currents, matter in some form is necessary for the process. 

Radiation is the name given to a process by which heat 
can travel without the aid of matter in any form. 

Radiant heat, as it is called, is a transverse wave motion 
exactly similar to that of light. In fact, radiant heat is 
sometimes spoken of as " invisible light." 

Heat rays may be reflected, refracted, transmitted and 
absorbed exactly like light rays. 

If we make a bar of iron white hot it emits light rays like 
any other source of light. It also emits heat rays, and although 
thev do not affect the eye in the same way as light, they can 
be felt. 

If the iron were heated to a temperature of 200 to 300 C. 
it would not emit any light rays, but the hand placed at a 
distance of a few inches would be able to detect the presence 
of heat rays. 

The heat which we feel from a fire is very largely due to 
radiation. Air, as we have seen, is a very bad conductor of 


heat, and the convection currents which are set in motion by 
the fire cause the hot gases to flow up the chimney. Hence 
very little heat reaches us from the fire by either conduction 
or convection. 

Experiment 63. 

Obtain a bright tin canister and make a mark on the 
inside about three-quarters of the distance up. 

Stand the tin on a block of wood and pour boiling water 
into the tin up to the mark. 

Place a thermometer, with its bulb in water, and record 
the temperature every half minute. Plot a graph showing 
the relation between time and temperature. 

A little thought should make it clear that the slope or 
steepness of the curve is an indication of the rate of cooling. 

The tin (and similar tins) should now be treated in different 
ways. One may be painted white, another black ; some may 
be lagged with paper of different thicknesses. 

Cooling curves should be obtained for each case and from 
these an opinion may be formed as to the best methods of 
encouraging or preventing radiation, according to the needs 
of the case. 

Experiment has shown that the rate of radiation depends 
upon : 

(1) The difference between the temperature of the radiating 
body and that of the surrounding bodies. 

(2) The area of the surface from which the radiation is 
taking place. 

(3) The nature of the radiating surface. 

Generally speaking, a smooth and shiny surface is a bad 
radiator, while a rough surface is a good radiator. White 
surfaces are not such good radiators as black, other things 
being equal. 

Experiment 64. 

Melt the wax in the test- tube used in Experiment 62 and 
place the bulb of a thermometer in the liquid. Having sup- 
ported the tube so that it is suspended in still air, take the 


temperature every half minute until the wax has all solidified. 
Express the results graphically. 

Fig. 97 shows the graph of a typical set of results. Observe 
that the left-hand portion of the curve is very steep. Here 
we have the hot liquid radiating its heat very quickly, since 
the difference of temperature between it and the surrounding 
objects is considerable. 

* & $ 3 













? 5 10 Id 20 21 


FIG. 97. 

The last portion of the curve is less steep. Here we have a 
solid whose temperature is not very much above that of its 
surroundings. Hence the cooling is taking place more slowly. 

In the middle of the curve we have a flat portion where the 
temperature remains constant for some minutes. 

Now although " cooling " ceased during this period, there 
is no reason to suppose that radiation ceased also. During 
the interval the paraffin was turning from the liquid to the 
solid state, and its latent heat had to be disposed of. 

Hence it is reasonable to assume that there was no inter- 
ruption in the process of radiation. But for the first seven 
or eight minutes the radiant heat was supplied by the sensible 


heat of the liquid paraffin. For the next eight or nine 
minutes the latent heat of the solidifying liquid was radiated. 

It is only after the whole of the liquid has solidified that 
cooling, as we generally understand it, takes place again. 
The sensible heat of the solid paraffin is then radiated. 

The temperature at which the graph flattens (in this case 
52 C.) is the melting point of the wax. 

Exercises 17. 

1. A rod 1" diameter is made partly of iron and partly of 
wood. If paper is wrapped tightly round the rod and a 
Bunsen flame is passed along the rod, state what will be 
observed. What does this experiment prove ? 

2. On a cold morning the fitter finds that his chisel appears 
to be much colder than his hammer handle. Give the 
explanation of this. 

3. An iron tube, 0*95" diameter and 15" long, had a number 
of brass fins fixed to it, 2" square. The tube was heated and 
then allowed to cool and the following readings of temperature 
in degrees F. were taken with time intervals of 1 minute. 
Plot time and temperature, heating and cooling curves. 

Heating curve, 1 minute intervals : 

61, 71, 79, 88, 99, 112, 124, 137, 150, 156, 164, 175, 184. 
Cooling curve, 1 minute intervals : 

150, 139, 129, 126, 122, 118, 114, 111, 108, 105, 102, 98, 
96, 94, 92, 90, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 76, 75. 

4. The following results give data for the heating and cooling 
curves of a saturated solution of salt and water. One 
minute interval was allowed between each reading. Plot time 
and temperature (Centigrade) for both the heating and 
cooling curves. 

Heating curve : 

14, 18, 28, 37-5, 47, 56, 65, 74, 81, 88, 91 '5, 106, 106 '5, 
107, 107. 


Cooling curve*. 

104, 102, 101, 100, 98, 96, 94, 92, 90, 88, 86, 84, 82, 80, 77, 
74, 72, 69-5, 68, 65'5, 63, 61, 59, 57'55, 53, 52, 51, 50. 

5. Steam has to be carried by piping a considerable distance 
away from a boiler. Explain why it is necessary to lag the 

6. Give sketches showing the methods of cooling the 
cylinders of motor cycle engines. On your sketches note 
whether the heat is conducted away by radiation, conduction, 
or convection. 

7. Explain why an eiderdown quilt makes a good bed- 
covering in winter. 

8. Apparatus for making ice cream consists of a thick wood 
bucket and an inner can of thin metal. Explain why. 

9. Explain the principles underlying the vacuum -flask. 

10. A room is heated by means of an open fire. Explain 
with the aid of sketches the various ways in which the heat 
is transmitted to the room. 

11. In order to economise fuel, food is sometimes cooked 
by means of a " hay box." Explain the underlying principles. 

12. A series of experiments was carried out by Brill to 
determine the value of various commercial coverings of 
steam pipes. A length of piping 60' long was used and the 
heat loss was determined by the condensed steam. The 
following are some of the results : 

Thickness of B.Th.U. radiated. 

Kind of covering. covering. per sq. ft. per min. 

Bare pipe 12'27 

Magnesia 1 "25" 1 -74 

Mineral wool 1'30" 1'29 

Fire felt 1'30" 2-28 

Hair felt 0'82" 1-91 

Explain how heat is lost from steam pipes and state what 

you can deduce from the above figures. 


13. A cooling curve was taken for melted tallow. Tempera- 
tures were measured in degrees Centigrade, and time intervals 
of one minute were allowed between each reading. Plot a 
graph of time and temperature. 

Cooling of melted liquid. 145, 140, 140, 134, 128-5, 122, 
116, 110, 105, 100, 95, 90, 85-5, 81, 76, 72, 68, 64, 60, 58, 56, 
52, 50, 47-5, 45, 42-5, 40, 38-5, 37 ; solidification starts ; 
35, 33-5, 33, 32, 31, 30*5; thin layer of solid crust ; 30, 30, 
29-5, 29, 28, 27 '5, 27, 27, 26*5. 

14. 'From the following figures, plot a graph, showing how 
the weight of water varies with temperature : 

Temperature. Weight in Ibs. per cub. ft. 

32 F. 62-42 

52 62-4 

62 62-36 

72 62-3 

82 62-21 

92 62-11 

102 62-0 

122 61-7 

142 61-34 

162 60-94 

182 60-5 

202 60-02 

212 59-76 

15. Give sketches of a small chamber, suitable for providing 
cold storage for a butcher. Explain the principles upon which 
it works. 

16. Tests were carried out with vessels of different metals to 
see how much steam per hour per square foot they were cap- 
able of condensing. The following are some results : 

Iron, 7-5. Brass, 12'5. Copper, 14'5. 
What can be deduced from these figures ? 

17. In a test to determine the specific heat of mild steel it 
was found that the water equivalent of the calorimeter was 
14'2 grammes, 121 '685 grammes of water at a temperature 


of 17C. were placed in the calorimeter and 38*85 grammes of 
mild steel at a temperature of 100 C. were dropped into the 
water and the water rose in temperature to 20 C. What is 
the specific heat of the mild steel ? 

18. Define the " thermal conductivity " of a substance. 

19. It is known that the freezing point of a salt solution is 
lower than that of pure water. Will this fact help you to 
explain why salt is often thrown on slippery frozen pavements ? 

20. Give reasons to show why lakes freeze more readily 
than oceans. 

21. Kent gives the following figures : 
Per cent, of salt by weight in water 

1 5 10 15 20 25 

Freezing point in degrees Fahrenheit 

31-8 25-4 18-6 12'2 6'86 1 

Plot a graph of per cent, of salt by weight and freezing 

22. What weight per cent, of salt will be found in sea water 
if it freezes at 27 F., assuming sea water to be a common 
salt solution ? 

23. When strong sulphuric acid is poured into a glass vessel 
containing water it often happens that the vessel breaks, and 
in any case a big rise in temperature is noted. Can you 
account for the glass cracking ? 



Energy. We have on one or two occasions referred to heat 
as a " form of energy." It is proposed here to consider this 
idea in a little more detail. 

It is a' matter of common experience that all the bodies 
with which we deal offer a resistance to motion. This re- 
sistance may be due to the gravitational force of the earth 
(if any vertical motion is attempted), or it may be due to 
friction between the body and its support. 

In any case the resistance is there, and we define " force " 
as that which tends to overcome resistance. 

It should be observed that a force may be applied without 
overcoming the resistance. For example, a force of 10 Ibs. 
applied to a railway truck would be insufficient to move it. 
That is, the resistance in this case would not be overcome. 

When resistance is overcome, we say that " work " is done, 
and the amount of work done is indicated by the product of 
the magnitude of the force and the distance through which 
it acts. 

Now " energy " is denned as that which is capable of 
doing work. There are a great many forms of energy. A 
moving weight is capable of doing work, and is said to possess 
" kinetic energy." A weight raised to a height above the 
earth's surface (e.g. a lake on a mountain side) possesses what 
is called " potential energy." 

Heat and electricity are forms of energy. The former 
can be made to do work by means of a heat engine, while 
the latter can operate an electric motor and thus do work. 

A piece of coal possesses what is called " chemical energy." 
When it burns it combines chemically with the oxygen of 
the air and heat is produced. 




Now any form of energy is (under suitable conditions) 
transformable into any other form. .But there is in every 
case what we may call a fixed rate of exchange. 

The amount of energy in a body is measured by the 
amount of work which it is capable of doing, and this is in 
all cases a definite and calculable quantity. 

The Mechanical Equivalent of Heat. It is a common saying 
that friction produces heat. It would be more correct to 
say that the overcoming of friction produces heat. Here 

FIG. 98. 

we have an example of kinetic energy being converted into 

The following is a description of an experiment in which 
the amount of friction overcome (i.e. the work done) is 
measured and also the number of units of heat produced. 

There are several forms of apparatus available. Fig. 98 
shows one of a very simple design. It consists of a hollow 
metal drum D which can be rotated by the handle H. 

Passing round the outside of the drum is a band of silk B. 
One end of this band is held down to the base of the instru- 


inent by means of a small spring balance and the other end 
supports a hanging weight-carrier. 

The drum is rotated so that it passes under the silk band 
in a direction passing from the weight carrier to the spring 
balance. A little thought will make it clear that the resist- 
ance offered at the surface of the drum is equal to the differ- 
ence between the weight supported and the reading of the 
spring balance. 

Engineering students will recognize this arrangement as a 
miniature rope-brake dynamometer, as used for the deter- 
mination of the horse-power of small engines. 

A known quantity of water is placed inside the drum and 
a specially bent thermometer is supported so that its bulb is 
always in the water. 

An example will illustrate the mode of operation. 

Ex. 27. 

Weight supported, 7*3 Ibs. 

Beading of spring balance, 0*7 Ibs. 

Diameter of drum, 6 inches. 

Number of revolutions, 107. 

Weight of drum, T27 Ibs. 

Weight of drum and water, 1 '955 Ibs. 

Temperature rise of water, 1*8 degrees Fahrenheit. 

It will be observed that the friction of the " brake " on the 
rim of the drum has produced heat and has raised the water 
(and the drum, too, of course) through 1 '8 degrees Fahrenheit. 

We will consider the heat first and will estimate it in British 
Thermal Units. 

Heat gained by water : 

= weight of water x degrees rise. 
= (1-955 1-27) Ibs. X 1*8 degrees. 
= '685 Ibs. X 1 '8 degrees. 
= 1-23 B.Th.U. 

Heat gained by drum : 

= Weight of drum X degrees rise X specific heat. 

^Now the drum was made of brass, which has a specific 
heat of 0-094), 


Hence heat gained by drum : 
= 1-27 Ibs. X 1-8 degrees X 0-094. 
=0-215 B.Th.U. 

Total heat produced = 1-23+0 '215. 
= 1-445 B.Th.U. 

Force exerted as friction by the brake =7 '3 0'7 lbs.= 
6-6 Ibs. 

And this force acted through a distance equal to thy 
circumference of the drum multiplied by the number of 
revolutions made. 

That is, the distance was : 
TTX 6x107 inches. 
= 168 feet. 

But work done is force x distance : 
=6-6 Ibs. X 168 feet. 
= 1108 ft.-lbs. 

Hence we see that : 

1-445 British Thermal Units of heat are equivalent to 
1108 ft.-lbs. of molar energy. 

/. 1 B.Th.U. = =767 ft.-lbs. 

This quantity is called the mechanical equivalent of heat. 
Of course the instrument described above does not give the 
most reliable results, as it lacks the somewhat complicated 
refinements of the more accurate instrument. The principle 
involved is, however, the same in all cases. 

The best results give a value to the mechanical equivalent 
of heat of 778 ft.-lbs. per B.Th.U. or 426 '9 kilogramme- 
metres per calorie. 

A kilogramme-metre is, of course, the work done by a force 
equal to the weight of 1 kilogramme acting through a distance 
of 1 metre. 

The Conservation of Energy. We 'have seen that 1 British 
Thermal Unit of heat on conversion into kinetic energy, 
produces 778 foot -Ibs. " These two amounts of different 


types of energy are interchangeable. It makes no difference 
whether heat is being converted into kinetic energy or vice 
versa, the equivalent is never varied. Nor is it affected by 
the mode of transformation. 

Of course our ignorance or carelessness may result in some 
of the energy reaching a destination which is not intended. 

For instance, in the experiment described in Example 27, 
it is quite possible that some of the heat produced by the 
brake was radiated into the air, and therefore failed to reach 
the water, where it would have been measured. 

Again, a steam engine converts heat into kinetic energy, 
but some of the heat produced in the furnace passes up the 
chimney-stack, and some remains in the exhaust steam, not 
to mention other means of escape. Hence only a portion 
(and in this case a very small portion) of the heat produced 
is actually converted into kinetic energy. 

In a modern steam engine about 10 per cent, of the heat 
produced in the furnace is converted into useful energy. 
The remaining 90 per cent, is there, but at present we do not 
know how to use it. 

The 10 per cent, in this case is called the " thermal effi- 
ciency " of the engine. Some types of heat engine (notably 
the internal combustion engine) have improved on this to 
some extent. The student should, however, grasp the fact 
that no amount of mechanical improvement or novel design 
can make an engine with a thermal efficiency of over 100 
per cent, and a great advance is necessary before this degree 
of efficiency is remotely approached. 

No mere mechanism can withdraw from a body more 
energy than it contains. 

Calorific Value of Fuels. If we take 1 Ib. of coal and burn 
it, heat is produced. This heat may be measured with a 
suitable calorimeter, and the result is known as the calorific 
value of the coal. 

Of course the numerical value depends upon the com- 
position of the coal, but for a given coal it is an absolutely 
fixed quantity. The number of thermal units produced 


does not depend upon the rapidity of burning, so long as the 
coal is completely burned. 

The following table gives the calorific values of a few 
common fuels : 


Calorific Value 
in B.Th.U's. per 
Ib. of fuel. 

Coal, Average composition 
Petroleum (commonly called 
paraffin) .... 
Hydrogen .... 
Carbon .... 



We sometimes see advertisements of patented prepara- 
tions which claim that when these are mixed with coal the 
latter will give twice as much heat as it otherwise would. 
No preparation can possibly produce this result, as it is 
contrary to the laws of nature. 

Such treatment may cause the coal to cake together in 
the grate and thus last longer, that is, to burn more slowly, 
but this is quite another thing. 

If 1 Ib. of coal is burnt in 10 minutes it is producing heat 
at the rate of 84,000 B.Th.U.'s per hour. 

If it takes half an hour to burn the heat is produced at 
the rate of 28,000 B.Th.U.'s per hour. 

In both cases, however, the 1 Ib. of coal produces its 14,000 
B.Th.U.'s. No more and no less. 

Electricity is a form of energy. If it is passed through a 
thin wire which offers a high resistance the wire becomes 
sensibly hot. Here we have electrical energy being con- 
verted into heat energy. 

It is found that 1048 watts of electricity flowing through 
a resistance for 1 second produce 1 British Thermal Unit 
of heat. 


Again, if electricity is passed through an electric motor, 
the latter revolves and may be made to do work. Here we 
have electrical energy being converted into kinetic energy. 

It is found that 1 watt of electricity flowing for 1 second, 
if wholly converted into kinetic energy, is capable of doing 
0-742 ft.-lbs. of work. 

Hence we have : 

1 watt-second of electricity is equivalent to 0'742 ft.- 
lbs. of work. 

.'. 1048 watt -seconds of electricity are equivalent to 
1048x0-742 ft.-lbs. of work. =778 ft.-lbs. 

But 1048 watt-seconds of electricity are equivalent to 
1 B.Th.U. of heat. 

/. 1 B.Th.U. of heat is equivalent to 778 ft.-lbs. of work. 

This will be recognised as the mechanical equivalent of 
heat already mentioned. 

We see, therefore, that energy may be transformed from 
one form to another. This may be done in a great variety 
of ways, but there is no known method whereby energy may 
be created out of nothing. Nor can we destroy energy, that 
is, turn it into nothing. 

Efficiency. Of course, we are seldom able to effect a 
complete transformation of the kind which we desire. 

Thus, if we deliver 1048 watt-seconds of electricity to 
any motor we shall get rather less than 778 ft.-lbs. of work 
from the motor, because some of the electricity will be 
transformed into heat, and in that case will not be available for 
transformation into kinetic energy also. 

Suppose in the case considered the motor is found" to 
give 645 ft.-lbs. of work. We see that electricity equivalent 
to 133 ft.-lbs. has been lost (not destroyed). Perhaps it has 
become heat. 

In any case the ratio of work obtained to work supplied is 


__- =0'83. -This is generally expressed as 83 per cent., and 


is called the " efficiency " of the motor. 

It indicates that, of the energy supplied to the motor, 
83 per cent, is converted into the form in which we require it. 


Exercises 18. 

1. Determine the work done in the following cases : 
Load in pounds . . 121'5 89 356'9 302-6 
Vertical distance moved in 

feet .... 29 32 52 9 '2 

2. A tug-of-war is in progress and for three minutes each 
side pulls with an exactly equal force of 495 pounds. How 
much work is done ? 

3. A stone weighs 225 pounds and a man pulls at the stone 
for 2 '4 minutes with a force of 110 pounds and fails to move 
it. How much work has the man done ? 

4. A man removes four shovels full of sand from the ground 
on to a platform 3' 6" high. By the aid of sketches and 
approximations show how much work the man does. 

5. A man is sandpapering a board which is 3' 6" long and 
12" wide. He rubs in one direction with a force of 23 pounds. 
If the rubber is 3" wide, how much work in foot pounds will 
the man do in order to rub each part of the board once ? 

6. A man is filing a piece of metal and on each forward 
stroke he applies a horizontal force of 15 pounds with a stroke 
of 6*5". If he makes 22 strokes, how much work in foot 
pounds will the man do ? 

7. A man stretches up his arms to hold a heavy weight. 
At the end of 5 minutes he feels tired. If the weight was 
37 pounds, state the work done in foot-pounds. Write a 
short essay showing the difference between the idea of " work " 
in a popular and a scientific sense. 

8. Illustrate with the aid of sketches a number of cases 
where work is being done by manual labour. In your sketch 
show the distance moved and the mass. Give approximate 
answers for the work done in 5 minutes hi each case. 

9. What do you understand by the term " work ? " 
State whether work is done in the following cases and give 
approximate answers : 


^1) boring a hole with a brace and bit. 

(2) carrying a load of 7 pounds 7 feet horizontally. 

(3) lifting 7 pounds vertically through 3 feet. 

(4) A string holding up a weight of 14 pounds. 

10. In a test of lifting tackle, the following results were 
obtained : 

Load lifted in pounds . .25 50 100 150 

Effort to move load . . 5J 7J 13J 19 

If th$ effort moves 20 times as fast as the load, determine 
the work done by the load and the effort when the load has 
moved 1'5 feet. Plot a graph of load and effort. 

11. In a test on pulley blocks it was found that the effort 
moved 4 times as fast as the load. Assuming that the load 
moved 4 feet, determine the foot-pounds of work done in 
moving the load and the effort through their respective 
distances. Plot curves of load and effort. The following are 
the figures required : 

Load in pounds 

5 10 15 20 25 30 

Effort in pounds 

3-8 6-3 8-9 11-0 12-6 14'7 

12. In a worm gearing it was found that the effort moved 
31 times as fast as the load. If the effort moved one foot, 
determine the number of foot-pounds of work utilized in 
moving the load through one foot. Plot a graph of load and 

Load in pounds 4 6 8 10 

Effort in pounds . 0'3 0'5 0'75 0'9 

13. The following calorific values in B.Th.U. per Ib. are 
given : Texas oil, 10,800 ; Trinidad crude, 10,200 ; shale 
oil, 10,120 ; heavy tar oil, 8,916. Convert these values into 
calories per gramme and equivalent foot-pounds of work. 

14. The following theoretical maximum combustion tem- 
peratures are given : Natural gas and air, 1,086 C. ; Thermit, 
2,694 C. ; oxyhydrogen flame, 3,190C. Convert these values 
into Fahrenheit degrees. 


15. In a determination of the electric horse -power of a 
motor the following observations were made : 

Volts . . . 490 490 490 490 

Amps. . . . 1-8 2-2 2 -6 2 '8 

Since amps. X volts = watts, determine the number of foot- 
pounds of work done per minute in each case. 

16. In a further test on the above motor, the brake horse- 
power was found to be as follows : 0'48, 0*73, and 0'95. 
If 33,000 ft. Ibs. per minute is equal to one horse-power, 
determine the foot-pounds of work per minute in each case. 

17. In a gas engine test the following heat distribution 
was obtained : 

Work obtained as indicated horse-power . 21*5% 

Lost in the jacket water . ... . 50*4 

Lost in the exhaust gas ..... 25 

Balance of the losses 3*1 


If the total indicated horse-power is 2 '3, determine these 
ratios as foot-pounds of work per minute. 

18. The average heating value of the following gases is 
given in B. Th.U. per cubic foot : Illuminating gas, 600 ; coke 
oven gas, 650 ; producer gas from coke, 135 ; blast furnace gas, 
100. Determine the equivalent values in foot-pounds of 

19. An engine is directly coupled to a generator and it is 
found that the engine develops 50 indicated horse-power. 
Neglecting all losses in engine and dynamo, determine the 
foot-pounds of work and the watts generated per second. If 
the work available from the engine is only 80 % of the I.H.P. 
and the efficiency of the generator is 95 %, how many watts 
are generated per second ? 

20. Wherever energy is used to produce motion, some of 
the energy is wasted. Write a short essay, illustrated by 
sketches, showing cases where energy is being wasted, and 
explain what becomes of the wasted work. 


21. An experiment was carried out to determine the mech- 
anical equivalent of heat and the following are the results : 

Weight of bottom cup of copper, 41 '15 grammes. 
Weight of top cup and water, 71*25 grammes. 
Weight of top cup, 55*4 grammes. 
Original temperature of the water, 17*5 C. 
Final temperature of water 30 '2 C. 

To cause this rise of temperature the apparatus made 
2,420 revolutions, a load of 0*155 Ibs. being applied to the rim 
of a 'wheel whose circumference is 800 millimetres. Deter- 
mine the mechanical equivalent of heat in ft.-lbs. per B.Th.U. 

22. In another experiment performed with a similar piece of 
apparatus the following results were obtained : 

Weight of two brass cones, 164*1 grammes. 
Weight of contained water, 21*35. 

Suspended mass, 300 grammes at the rim of a wheel whose 
circumference was 78*4 centimetres. The total number of 
revolutions was 1,852 and the temperature rise was 24 C C. 
Determine the mechanical equivalent of heat. 

23. Zinc is dissolved by the action of dilute sulphuric acid, 
and the final product is zinc sulphate, hydrogen, and water. 
The chemical energy of the products is less than that of 
the original substances. In carrying out this experiment 
explain what you have noticed. Where has the lost energy 
gone ? 


Exercises 1. 
1. 26-19,52-39, 77 -79 mm. 

2. r, A', 121*. 

3. 3-609, 4-921, 12-795 ft. 

4. 0-186,0-403, l-506sq. in. 

5. 0-4989, 1-1340, 2-1319 kgms. 

6. 11-24, 14-33, 16-98 Ibs. 

7. 48, 89-6, 118-4 pdls. 

8. 88 ft. per sec., 2680 cms. per sec. 

9. 1026 ft. per min. 

12. 2640 ft. per min. ; 80400 cms. per min. 

13. 45 ft. per sec. 

14. 60 miles per hr. per hr. 
0*024 ft. per sec. per sec. 
0*74 cms. per sec. per sec. 

15. 3*09 X 10 7 dynes per sq. cm. 

16. 6*9 X 10 6 dynes per sq. cm. 

17. 32 ft.-lbs., 1025 ft.-pdls., 4'3x 10 8 ergs. 

18. 55-43, 33-43. 

Exercises 2. 

1. 6-859, 13-824, 79'507, 0'9112 c.c. 

2. 0-007854, 1-131, 8'55, 15'90 cub. in. 

3. 0-000523, 0-0335, 8-181, 344-8 cub. in. 

4. 1-45. 

5. 0-283 Ibs. per cub. in., 49'17 c.c., 385'5 gms., 7'85. 

6. 0-283, 0-284, 0'283 Ibs. per cub. in. 

7. 0-284, 0-283, 0'283 Ibs. per cub. in. 

8. 0-361, 0-361, 0-361 Ibs. per cub. in. 

9. 1, 2-59, 6-68, 9'82 gms. per c.c. 



10. 62 '5 Ibs. per cub. ft., 1 gm. per c.c. 

11. 2.57, 8-5, 8-9, 7*7. 

12. 2-5, 2-6, 19-26, 7 -69, 10'5. 

13. 0-283, 0-277, 0'314, 0*297 Ibs. per cub. in, 

14. 0-67, 0-75, 0-72, 0'74, 0'66. 

15. 96, 115-3, 49-5, 55 '6 Ibs. 

Exercises 3. 

1. 1*14 gms. per c.c. 

2. 8-55 

3. 0-859 

4. 13-59 

5. 13-47 

6. 13-61 

7. 7-82 

8. 0-52 

9. 0-818, 1-22,0-87. 

10. 1'06 gms. per c.c. 

11. 0-8. 

12. 1-34. 

13. 0-78. 

15. 2*41 gms. per c.c. 

Exercises 4. 

1. I'll Ibs. per sq. in. 

2. 0-039 Ibs. per sq. in. 

3. 4*3 Ibs. per sq. in. 

4. 14 '5, 2-2 Ibs. per sq. in. 

5. 26-5". 

6. 0-0141, 0-0154, 0-0159 Ibs. per sq. in. 

7. 1798. 

9. 0*014 Ibs. per sq. in. 

15. 0-144". 

16. Less than 46 Ibs. 

19. 0-491. 

20. 29-9, 26-55, 23'45, 20'6, 18'15 inches. 

21. 559, 564, 569, 574 mm. 
91-64, 91-87, 92-11, 92*34 C. 

22. 2-95, 3-55, 4'14. 4'73, 5'32 inches. 


23. 2-73 cub. in. 

24. 83-4 cub. in. 

25. 14-22, 14-35, 14'67, 14'80, 14*90, 15'08 Ibs. per sq. in. 

26. 0-99909, 0'99989, 1 '00005 gms. per c.c. 

27. 50 cub. ft. 

28. 26-1 ft. 

Exercises 5. 

4. 0-65 sec. 14. 2 sees. 

5. 1-25 sec. 15. 9'8". 

6. 0-55 sec. 16. 0'073" 

Exercises 6. 

9. 329, 1493, 251, 507, 5340, 3165, 3352 metres per sec. 

12. C'. 

13. Middle C. 14*9 revs, per sec. 

\ 4 _ 83 - 34 list ' -' i 313 is 9 

*" ^.->i;> 45 ">';> 6" > 2> 128 > 8 "* 

15. E, G. C'. 

16. C,,,, A //5 E. 
18. Middle C. 

20. 200 feet per sec. 

Exercises 8. 
1. 1, 2, 4. 

4. 41 -9 ft. 

5. 1620 yds. 

6. 0-416, 14-7 sees. 

16. 336-5, 346, 349 metres per sec. 

Exercises 9. 

3. 0-0000004861, 0*0000004308, 0-0000005167, 0*000000382 


4. 0-0000486, 0-00008, 0'0002, 0'00025 cm. 
6. 8*3 minutes. 

8. Circle of 3" diameter. 
Vertical line 3" long. 
11. Y'/' or r. 
15. 32-7,26-7,0-9, 20 '4, 27 -6. 

17. 0-093. 


Exercises 10. 

7. 60. 

8. 0-083, 0*125, 0-127, 0'318. 

14. 40. 

Exercises 11. 

9. 140300, 137200, 124400, 111700 miles per second. 

15. Hft. 

16. 2*46, 3'8, 4, 3'57, 4'5 inches. 

Exercises 12. 

5. 150, 150, 160, 200, 360 mm. 

6. 150, 150, 160, 200 mm. 

7. 7-3, 7-1, 4-88, 20*9, 21'2, 15'7, 14*2 cms. 
10. 19-275 cms. 22-742 cms. 

16. 15 cms. 
18. 25-6 cms. 

Exercises 13. 

13. 1-7C. 

14. 36-9 C., 20 C. 

15. 1094, 1148, 1310, 1472, 1687, 1742, 1994, 2040, 2282 F. 
863, 893, 983, 1073, 1193, 1223, 1363, 1423, 1523 A. 

16. 1051, 1175, 1375, 1549, 1650, 1726, 1771, 2201 F. 

17. 156, 430-5, 218, 61. 

18. Pressures : 0'95, 1, 2'04, 6 ! 1, 6*8 atmospheres. 
Temperatures: 98 -6, 100, 121 '4, 160'3, 164'4, C. 

19. -188 F., -182 F., -229F. 
706, 821, 494 Ibs. per sq. in. 

21. 224 C., 299 C., 316 C., 427 C. 

Exercises 14. 

1. 8-0005", 8-001". 

2. 4-9998", 5-0003. 

3. 2-0004, 4-0009 6'0013, 8-0017. 


4. 6-012". 24. 1-45". 

5. 9-9986 ft, 26. 0-037 cub. in. 

6. 0-26" 29. ,V 

7. 0-0018 inch per foot 32. 0-0000186. 

8. 0, 0-0002, 0-00045, O'OOOT. 33. 0-000167. 

9. 0-00084". 34. 0'004. 

10. 0-000347". 35. 0'00365. 

11. 0-013 per cent. 41. 165 c.c. 
15. 0-136". 42. '29-5 c.c. 
21. 0-148". 43. 140. 

Exercises 15. 

1. 2-72 xlO 5 , 1-37X10 6 , 1-45X10 6 , 1 93 XlO 8 calories. 
272, 1370, 1450, 1930 grand calories. 

2. 1080, 5440, 5740, 7650 B.Th.U. 

5. 4230, 2200, 2490 B.Th.U. 

6. 5660 B.Th.U. per hour. 

7. 3-94, 3-33, 3 -05, 2 -83 B.Th.U. 
993, 840, 768, 713 calories. 

8. 5-11 X 10 6 , 5-01 X 10 6 , 5-03 X 10 6 , 4'69x 10 3 calories per Ib, 

9. l-24x 10 6 , 4-18X 10 6 , 2-1 x 10*, 3'21 x 10 6 B.Th.U. per Ib, 
10. 0-168, 0-0818, 0-0226, 0-186, 0'045, 0'069. 

12. 0-34 Ib. 3 B.Th.U. 

13. 350-97 calories. 

14. 0-14. 

15. 11-4, 11-3. 
16 0-1. 

18. 39-5 C. 

19. 38,100 B.Th.U. 

20. 282, 515, 244 B.Th.U. 

21. 183-5 Ib.-deg. Cent, units. 

22. 11200, 14540.. 13600 B.Th.U. 
24. 35-3 Ibs. per sq. in. 

Exercises 16. 
6. 977 B.Th.U. per Ib. 7. 79-1. 


8. 518. 

16. 1405, 875, 106 B.Th.U. 
0-019, 0-039, 0-014, 

17. 6-74 xlO 6 B.Th.U., 1-6 X 10 7 B.Th.U. 

22. 0-604, 0-864. 

23. 9176 B.Th.U. 

24. 63 C. 

' Exercises 17. 

17. 0-13. 22. 4 per cent. 

Exercises 18. 

1. 3525, 2848, 18550, 2780. 

5. 322 ft.-lbs. 

6. 178-7 ft.-lbs. 

10. Work on load : 37*, 75, 150, 225 ft.-lbs. 
Work by effort : 167J, 225, 397^, 570 ft.-lbs. 

11. Work on load : 20, 40, 60, 80, 100, 120 ft.-lbs. 
Work by effort : 61, 101, 142, 176, 202, 235 ft.-lbs. 

12. 0-129, 0-193, 0-258, 0'328 ft.-lbs. 

13. 2 -72 x 10 6 , 2 -57 X 10 6 , 2 -6 X 10 6 , 2 '25 X 10 8 calories. 
8 -4 XlO 8 , 7-94X10 6 , 7'88xl0 8 , 6 '94 x 10 6 B.Th.U. 

14. 1987 F., 4881 F., 5774 F. 

15. 39300, 48000, 56800, 61100 ft.-lbs. 

16. 15840, 24090, 31350 ft.-lbs. 

17. 16330, 38300, 18980, 2290 ft.-lbs. per minute. 

18. 466800, 505700, 105030, 77800 ft.-lbs. 

19. 27500 ft.-lbs. per sec. 
37100 watts, 28150 watts. 

21. 784 ft.-lbs. per B.Th.U. 

22. 0'36 kgm. -metres per calorie. 


Absolute temperature, 154 
Acceleration, 13 
Adiabatic expansion, 189 
Amplitude, 72 
Atmospheric pressure, 46 

Balance, the, 22 

Balance wheel, 70 

Barometer, 47 

Beats, 83 

Boyle's Law, 53 

British Thermal Unit, 181 

Calorie, 180 

Calorific value of fuels, 226 
Candle power, 109 
Centigrade temperature, 151 
Coefficient of expansion, 162 
Conduction, 210 
Conservation of energy, 225 
Convection currents, 212 

Density, 21 
Diffusion, 57 

Efficiency, 228 
Energy, 14, 222 
Expansion, 161 

Fahrenheit temperature, 151 
Floating bodies, 32 
Fluid pressure, 44 
Fluids, 28 

Focus, 138 
Force, 13 
Frequency, 73 
Fusion, 198 

Gas equation, 173 
Gases, 46 

Hare's apparatus, 38 
Harmonics, 89 
Heat, 146 
Hydrometer, 33 

Images, 132 
Inverse squares, 107 
Isothermal expansion, 189 

Latent heat, 195 
Length, 11 
Lenses, 137 
Level, 30 
Light, 104 
Loops, 88 

Magnifying glass, 141 

Mass, 11 

Matter, 20 

Mechanical equivalent of 

heat, 223 
Melting, 200 
Mercury, precautions in the 

use of, 48 
Mirrors, 112, 133 


240 INDEX 

Monochord, 80 
Musical scale, 81 

Nodes, 88 
Organ pipes, 100 

Pendulum, 67 
Periodic motion, 64 
Photometer, 106 
Prisms, 125 
Pumps, 54 

Radiation, 215 
Reeds, 94 
Reflection, 112 
Refraction, 121 
Refractive index, 123 
Resonance, 89 

Shadows, 105 

Simple harmonic motion, 66 

Sound, 77 

Specific gravity, 20 
Specific gravity bottle, 28 
Specific heat, 180 t 
Spherometer, 135 
Syphon, 39 

Temperature, 147 

Thermometer, 148 

Time, 11 

Total internal reflection, 126 

Tuning forks, 97 

Units, 11 

Velocity, 13 
Vibration, 77 
Virtual images, 137 

Water equivalent, 181 
Wave length, 72 
Wave motion, 70 
Weight, 12 
Work, 14 






University of Toronto 








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Under Pat. "Ref. Index File"